Aimathic
Login | English | Deutsch

Free math worksheets

Build your own math worksheets from 30,000+ problems for grades 3 to 12, from fractions to AP Calculus. Every problem comes with step-by-step solutions.

Estimate limits from tables

Click problems to add them to your worksheet.

54249012
The table gives values of \(f(x)\) near \(x=2\). <table><tr><th>\(x\)</th><td>\(1.9\)</td><td>\(1.99\)</td><td>\(1.999\)</td><td>\(2.001\)</td><td>\(2.01\)</td><td>\(2.1\)</td></tr><tr><th>\(f(x)\)</th><td>\(4.81\)</td><td>\(4.981\)</td><td>\(4.9981\)</td><td>\(5.0019\)</td><td>\(5.019\)</td><td>\(5.19\)</td></tr></table> Estimate \(\lim_{x\to2}f(x)\).

Hints

- Compare the outputs for x-values closest to \(2\). - Look at the left and right sides separately first. - A two-sided estimate is supported when both sides approach the same value.

Solution

1. For x-values less than \(2\), the outputs approach \(5\). 2. For x-values greater than \(2\), the outputs also approach \(5\). 3. Therefore \(\lim_{x\to2}f(x)\approx 5\).

Answer

\(\lim_{x\to2}f(x)\approx 5\).
54250712
The table shows the behavior of \(G(x)\) for larger and larger positive inputs. <table><tr><th>\(x\)</th><td>\(10\)</td><td>\(100\)</td><td>\(1000\)</td><td>\(10{,}000\)</td></tr><tr><th>\(G(x)\)</th><td>\(4\)</td><td>\(16\)</td><td>\(64\)</td><td>\(256\)</td></tr></table> Does the table support a finite value for \(\lim_{x\to\infty}G(x)\)? Describe the trend instead.

Hints

- Look for whether the output row is leveling off. - Compare the size of successive outputs as the inputs grow. - Distinguish approaching a finite number from growing without bound.

Solution

1. As the inputs increase, the outputs grow from \(4\) to \(256\) rather than leveling off. 2. The displayed values provide no evidence of approaching a finite number. 3. The table suggests unbounded positive growth as \(x\to\infty\), so no finite limit is supported.

Answer

No finite limit is supported. The table suggests \(G(x)\to+\infty\) as \(x\to\infty\).
54250912
The table shows values of \(J(x)\) for increasingly negative inputs. <table><tr><th>\(x\)</th><td>\(-10\)</td><td>\(-100\)</td><td>\(-1000\)</td><td>\(-10{,}000\)</td></tr><tr><th>\(J(x)\)</th><td>\(-0.25\)</td><td>\(-0.025\)</td><td>\(-0.0025\)</td><td>\(-0.00025\)</td></tr></table> Estimate \(\lim_{x\to-\infty}J(x)\). From which side of the limiting y-value do the table values approach?

Hints

- Compare the magnitudes of the outputs as the inputs decrease without bound. - Notice the sign of every output. - A limit of zero can be approached entirely from negative values.

Solution

1. As \(x\) becomes more negative, the outputs become closer to \(0\) in magnitude. 2. Every displayed output is negative, so the values approach \(0\) from below. 3. Therefore \(\lim_{x\to-\infty}J(x)\approx0\).

Answer

\(\lim_{x\to-\infty}J(x)\approx0\), approached from below.
54249112
Use the table to determine whether \(\lim_{x\to-1}g(x)\) exists. <table><tr><th>\(x\)</th><td>\(-1.1\)</td><td>\(-1.01\)</td><td>\(-1.001\)</td><td>\(-0.999\)</td><td>\(-0.99\)</td><td>\(-0.9\)</td></tr><tr><th>\(g(x)\)</th><td>\(2.8\)</td><td>\(2.98\)</td><td>\(2.998\)</td><td>\(4.002\)</td><td>\(4.02\)</td><td>\(4.2\)</td></tr></table>

Hints

- Split the table at \(x=-1\). - Estimate the limiting output on each side. - Compare the two one-sided estimates before deciding on a two-sided limit.

Solution

1. From the left of \(-1\), the values approach \(3\). 2. From the right of \(-1\), the values approach \(4\). 3. Since the one-sided estimates disagree, \(\lim_{x\to-1}g(x)\) does not exist.

Answer

The limit does not exist; the left-hand values approach \(3\) and the right-hand values approach \(4\).
54249212
A table samples \(h(x)\) near \(x=0\). <table><tr><th>\(x\)</th><td>\(-0.1\)</td><td>\(-0.01\)</td><td>\(-0.001\)</td><td>\(0.001\)</td><td>\(0.01\)</td><td>\(0.1\)</td></tr><tr><th>\(h(x)\)</th><td>\(10\)</td><td>\(100\)</td><td>\(1000\)</td><td>\(1000\)</td><td>\(100\)</td><td>\(10\)</td></tr></table> Does the table support a finite value for \(\lim_{x\to0}h(x)\)? Describe the observed behavior.

Hints

- Compare output size with distance from \(0\). - Ask whether the outputs are stabilizing near a finite number. - Large growth on both sides signals unbounded behavior rather than a finite estimate.

Solution

1. On both sides of \(0\), the outputs become much larger as \(x\) gets closer to \(0\). 2. The values do not settle near any finite number. 3. The table therefore does not support a finite limit; it suggests unbounded positive behavior near \(0\).

Answer

No finite limit is supported. The table suggests unbounded positive behavior as \(x\to0\) from both sides.
54249312
A table of values near \(x=3\) is shown, and separately you are told that \(q(3)=10\). <table><tr><th>\(x\)</th><td>\(2.9\)</td><td>\(2.99\)</td><td>\(2.999\)</td><td>\(3.001\)</td><td>\(3.01\)</td><td>\(3.1\)</td></tr><tr><th>\(q(x)\)</th><td>\(6.8\)</td><td>\(6.98\)</td><td>\(6.998\)</td><td>\(7.002\)</td><td>\(7.02\)</td><td>\(7.2\)</td></tr></table> Estimate \(\lim_{x\to3}q(x)\). Explain the role of \(q(3)=10\).

Hints

- Use the rows near but not equal to \(x=3\). - Compare the values from the left and right. - Keep the function value at the point separate from the nearby trend.

Solution

1. The table values from both sides approach \(7\). 2. Therefore \(\lim_{x\to3}q(x)\approx7\). 3. The separate value \(q(3)=10\) does not affect the estimate because the limit concerns nearby inputs rather than the single input \(3\).

Answer

\(\lim_{x\to3}q(x)\approx7\). The value \(q(3)=10\) does not change the limit.
54249412
The table lists values of a function \(m\) near \(x=2.5\). The entries are not ordered by distance from \(2.5\). <table><tr><th>\(x\)</th><td>\(2.2\)</td><td>\(2.49\)</td><td>\(2.7\)</td><td>\(2.501\)</td><td>\(2.51\)</td><td>\(2.499\)</td></tr><tr><th>\(m(x)\)</th><td>\(3.10\)</td><td>\(3.245\)</td><td>\(3.35\)</td><td>\(3.2505\)</td><td>\(3.255\)</td><td>\(3.2495\)</td></tr></table> Estimate \(\lim_{x\to2.5}m(x)\), and identify which pair of table entries gives the strongest evidence for your estimate.

Hints

- First compare how close each input is to \(2.5\). - Use evidence from both sides of the target input. - The most informative rows are usually those nearest the approach value.

Solution

1. The inputs closest to \(2.5\) are \(2.499\) and \(2.501\), one on each side. 2. Their outputs are \(3.2495\) and \(3.2505\), which bracket \(3.25\) very closely. 3. Therefore \(\lim_{x\to2.5}m(x)\approx3.25\).

Answer

\(\lim_{x\to2.5}m(x)\approx3.25\). The strongest evidence comes from \(x=2.499\) and \(x=2.501\).
54249512
A function \(r\) is sampled near \(x=-2\). <table><tr><th>\(x\)</th><td>\(-2.2\)</td><td>\(-2.02\)</td><td>\(-2.002\)</td><td>\(-1.998\)</td><td>\(-1.98\)</td><td>\(-1.8\)</td></tr><tr><th>\(r(x)\)</th><td>\(-0.18\)</td><td>\(0.024\)</td><td>\(-0.003\)</td><td>\(0.004\)</td><td>\(-0.021\)</td><td>\(0.16\)</td></tr></table> The outputs do not approach the limit monotonically. Estimate \(\lim_{x\to-2}r(x)\) anyway, and explain what feature of the table matters more than monotonicity.

Hints

- Focus on distance from the target input rather than whether the outputs always increase or decrease. - Compare the magnitudes of the outputs nearest \(-2\). - Check whether both sides are clustering around one value.

Solution

1. The outputs nearest \(x=-2\) are \(-0.003\) and \(0.004\), both close to \(0\). 2. Values on both sides become small in magnitude as the inputs move closer to \(-2\), even though their signs vary. 3. Therefore \(\lim_{x\to-2}r(x)\approx0\).

Answer

\(\lim_{x\to-2}r(x)\approx0\). What matters is that values from both sides get close to the same number as \(x\) gets close to \(-2\); they do not need to move toward it monotonically.
54249712
The table gives values of \(v(x)\) for increasingly large positive inputs. <table><tr><th>\(x\)</th><td>\(10\)</td><td>\(100\)</td><td>\(1000\)</td><td>\(10{,}000\)</td></tr><tr><th>\(v(x)\)</th><td>\(1.7000\)</td><td>\(1.9700\)</td><td>\(1.9970\)</td><td>\(1.9997\)</td></tr></table> Use the table to estimate \(\lim_{x\to\infty}v(x)\). State one limitation of using only this finite table as evidence for an infinite-input limit.

Hints

- Read the output trend as the inputs become much larger. - Look for a value the outputs are getting closer to. - Distinguish numerical evidence from a proof about all large inputs.

Solution

1. As the inputs increase by powers of \(10\), the outputs move closer to \(2\): \(1.7000, 1.9700, 1.9970, 1.9997\). 2. The table therefore supports the estimate \(\lim_{x\to\infty}v(x)\approx2\). 3. A finite table cannot by itself prove what happens for all sufficiently large inputs; it only provides numerical evidence for the trend.

Answer

\(\lim_{x\to\infty}v(x)\approx2\). The table supports the estimate but does not prove the behavior for arbitrarily large \(x\).
54249812
A table contains only values with \(x<6\). <table><tr><th>\(x\)</th><td>\(5\)</td><td>\(5.8\)</td><td>\(5.98\)</td><td>\(5.998\)</td></tr><tr><th>\(h(x)\)</th><td>\(8.4\)</td><td>\(8.88\)</td><td>\(8.988\)</td><td>\(8.9988\)</td></tr></table> What limit can reasonably be estimated from this table? Can the table alone determine \(\lim_{x\to6}h(x)\)? Explain.

Hints

- Check whether every listed input lies on the same side of \(6\). - Identify which one-sided limit the data address. - Ask what additional side would be needed for a two-sided conclusion.

Solution

1. All listed inputs approach \(6\) from the left, and the outputs approach \(9\). 2. Thus the table supports \(\lim_{x\to6^-}h(x)\approx9\). 3. The table provides no values for \(x>6\), so it gives no evidence about the right-hand limit. Therefore it cannot by itself determine the two-sided limit.

Answer

The table supports \(\lim_{x\to6^-}h(x)\approx9\). It does not by itself determine \(\lim_{x\to6}h(x)\) because no right-side data are given.
54249912
Two tables sample the same function \(F\) near \(x=4\). <table><tr><th colspan="3">Coarser sample</th></tr><tr><th>\(x\)</th><td>\(3.5\)</td><td>\(4.5\)</td></tr><tr><th>\(F(x)\)</th><td>\(9.4\)</td><td>\(10.6\)</td></tr></table> <table><tr><th colspan="5">Finer sample</th></tr><tr><th>\(x\)</th><td>\(3.99\)</td><td>\(3.999\)</td><td>\(4.001\)</td><td>\(4.01\)</td></tr><tr><th>\(F(x)\)</th><td>\(9.988\)</td><td>\(9.9988\)</td><td>\(10.0012\)</td><td>\(10.012\)</td></tr></table> Estimate \(\lim_{x\to4}F(x)\). Which table gives the more reliable estimate, and why?

Hints

- Compare how close each table's inputs are to \(4\). - Look for a common value suggested by both sides. - Finer sampling near the target usually gives better evidence about a limit.

Solution

1. The coarse table suggests a value near \(10\), since \(9.4\) and \(10.6\) lie on opposite sides of it. 2. The finer table gives outputs extremely close to \(10\) for inputs much closer to \(4\) from both sides. 3. Thus \(\lim_{x\to4}F(x)\approx10\), with the finer table providing stronger numerical evidence.

Answer

\(\lim_{x\to4}F(x)\approx10\). The finer table is more reliable because its inputs lie much closer to \(4\) on both sides.
54250012
The table samples a function \(Q\) only for inputs less than \(-4\). <table><tr><th>\(x\)</th><td>\(-5\)</td><td>\(-4.1\)</td><td>\(-4.01\)</td><td>\(-4.001\)</td></tr><tr><th>\(Q(x)\)</th><td>\(8\)</td><td>\(80\)</td><td>\(800\)</td><td>\(8000\)</td></tr></table> Describe the behavior suggested by the table as \(x\to-4^-\). Does the table support a finite left-hand limit?

Hints

- Track the output size as the inputs move closer to \(-4\). - Ask whether the outputs are stabilizing near a finite number. - Keep the one-sided direction in view when stating the behavior.

Solution

1. As the inputs approach \(-4\) from the left, the outputs increase from \(8\) to \(8000\). 2. The values grow rather than settling near a finite number. 3. The table suggests \(Q(x)\to+\infty\) as \(x\to-4^-\), so it does not support a finite left-hand limit.

Answer

The table suggests \(Q(x)\to+\infty\) as \(x\to-4^-\). No finite left-hand limit is supported.
54250112
The table samples \(f(x)\) closer and closer to \(x=0\). <table><tr><th>\(x\)</th><td>\(-0.1\)</td><td>\(-0.01\)</td><td>\(-0.001\)</td><td>\(0.001\)</td><td>\(0.01\)</td><td>\(0.1\)</td></tr><tr><th>\(f(x)\)</th><td>\(1\)</td><td>\(-1\)</td><td>\(1\)</td><td>\(-1\)</td><td>\(1\)</td><td>\(-1\)</td></tr></table> Does this table support a single value for \(\lim_{x\to0}f(x)\)? Explain what feature of the data prevents a limit estimate.

Hints

- Ignore whether the inputs are positive or negative at first and focus on the output pattern near \(0\). - Ask whether the outputs settle into a smaller neighborhood around one number. - Persistent oscillation is different from ordinary numerical noise that shrinks near the target.

Solution

1. The displayed inputs get progressively closer to \(0\), but the outputs continue to alternate between \(1\) and \(-1\). 2. The values do not cluster around one number as the inputs approach \(0\). 3. Therefore the table does not support a single two-sided limit value.

Answer

No. The outputs keep switching between \(1\) and \(-1\) instead of approaching one value as \(x\to0\).
54250312
The table samples a function \(A\) at larger and larger positive inputs. <table><tr><th>\(x\)</th><td>\(20\)</td><td>\(200\)</td><td>\(2000\)</td><td>\(20{,}000\)</td></tr><tr><th>\(A(x)\)</th><td>\(5.20\)</td><td>\(4.95\)</td><td>\(5.012\)</td><td>\(4.997\)</td></tr></table> Estimate \(\lim_{x\to\infty}A(x)\). Why does alternating above and below the estimated limit not by itself prevent convergence?

Hints

- Compare each output with \(5\). - Focus on the size of the deviation, not only its sign. - A convergent sequence of sampled values need not approach from only one side.

Solution

1. The outputs alternate around \(5\): first above, then below, then above, then below. 2. Their deviations from \(5\) become small as the inputs grow. 3. The table therefore supports \(\lim_{x\to\infty}A(x)\approx5\); crossing the limiting value is compatible with convergence when the values get progressively closer to it.

Answer

\(\lim_{x\to\infty}A(x)\approx5\). Alternating sides does not prevent convergence when the outputs become closer and closer to the same value.
54250412
The x-values in this table are not equally spaced around \(10\). <table><tr><th>\(x\)</th><td>\(9.7\)</td><td>\(9.97\)</td><td>\(9.997\)</td><td>\(10.0005\)</td><td>\(10.02\)</td><td>\(10.4\)</td></tr><tr><th>\(C(x)\)</th><td>\(1.85\)</td><td>\(1.985\)</td><td>\(1.9985\)</td><td>\(2.00025\)</td><td>\(2.01\)</td><td>\(2.20\)</td></tr></table> Estimate \(\lim_{x\to10}C(x)\). Explain why equal spacing on the two sides of \(10\) is not required for a numerical limit estimate.

Hints

- Identify the nearest inputs on each side of \(10\). - Compare their outputs rather than trying to pair equal distances. - A limit concerns approach from both sides, not symmetry of the table.

Solution

1. Inputs on both sides of \(10\) get very close to \(10\). 2. The corresponding nearby outputs, such as \(1.9985\) and \(2.00025\), are close to \(2\). 3. Thus \(\lim_{x\to10}C(x)\approx2\). Equal spacing is unnecessary because the key question is whether values from each side approach the same output.

Answer

\(\lim_{x\to10}C(x)\approx2\). The two sides need not use matching distances from \(10\); they only need to provide values increasingly close to the target from both directions.
54250512
A single data set samples the same function near two different inputs. <table><tr><th>\(x\)</th><td>\(-1.01\)</td><td>\(-1.001\)</td><td>\(-0.999\)</td><td>\(-0.99\)</td><td>\(1.99\)</td><td>\(1.999\)</td><td>\(2.001\)</td><td>\(2.01\)</td></tr><tr><th>\(E(x)\)</th><td>\(3.96\)</td><td>\(3.996\)</td><td>\(4.004\)</td><td>\(4.04\)</td><td>\(-2.97\)</td><td>\(-2.997\)</td><td>\(-3.003\)</td><td>\(-3.03\)</td></tr></table> Estimate \(\lim_{x\to-1}E(x)\) and \(\lim_{x\to2}E(x)\). Explain why values near the other target should be ignored for each estimate.

Hints

- Separate the table into the cluster near \(-1\) and the cluster near \(2\). - For each target, use values from both sides of that target only. - Do not mix data from neighborhoods centered at different inputs.

Solution

1. Near \(x=-1\), values from both sides approach \(4\), so \(\lim_{x\to-1}E(x)\approx4\). 2. Near \(x=2\), values from both sides approach \(-3\), so \(\lim_{x\to2}E(x)\approx-3\). 3. A limit at one target depends on behavior near that target; rows clustered around the other input are not relevant evidence.

Answer

\(\lim_{x\to-1}E(x)\approx4\) and \(\lim_{x\to2}E(x)\approx-3\). For each estimate, only values near that approach point are relevant; values clustered near the other target should be ignored.
54250612
Only a coarse numerical sample is available near \(x=-5\). <table><tr><th>\(x\)</th><td>\(-5.4\)</td><td>\(-5.2\)</td><td>\(-4.8\)</td><td>\(-4.6\)</td></tr><tr><th>\(F(x)\)</th><td>\(5.6\)</td><td>\(5.8\)</td><td>\(6.2\)</td><td>\(6.4\)</td></tr></table> Which estimate is best supported for \(\lim_{x\to-5}F(x)\): \(5\), \(6\), or \(7\)? Explain why the estimate should be viewed as less precise than one based on inputs much closer to \(-5\).

Hints

- Compare the values on the two sides of \(-5\). - Look for the choice around which the outputs are centered. - Consider how close the nearest input actually is to the target.

Solution

1. The values below \(-5\) are below \(6\), while the values above \(-5\) are above \(6\). 2. The pattern is centered near \(6\), so \(6\) is the best-supported estimate among the choices. 3. Because the closest inputs are still \(0.2\) away from \(-5\), the table gives only coarse evidence and does not justify a highly precise estimate.

Answer

\(6\) is the best-supported estimate. The data are relatively far from \(-5\), so the table supports only a coarse approximation.
54250812
The table gives very small output values near \(x=4\). <table><tr><th>\(x\)</th><td>\(3.9\)</td><td>\(3.99\)</td><td>\(3.999\)</td><td>\(4.001\)</td><td>\(4.01\)</td><td>\(4.1\)</td></tr><tr><th>\(H(x)\)</th><td>\(0.0018\)</td><td>\(0.00198\)</td><td>\(0.001998\)</td><td>\(0.002002\)</td><td>\(0.00202\)</td><td>\(0.0022\)</td></tr></table> Estimate \(\lim_{x\to4}H(x)\) to four decimal places. Why would reporting the answer as \(0\) lose important information?

Hints

- Focus on the entries paired with inputs closest to \(4\). - Keep enough decimal places to distinguish a small nonzero value from zero. - Use both sides before applying the requested rounding.

Solution

1. The values nearest \(x=4\) are \(0.001998\) and \(0.002002\). 2. These values approach \(0.002\) from opposite sides. 3. To four decimal places, \(\lim_{x\to4}H(x)\approx0.0020\). Reporting \(0\) would discard the nonzero limiting value indicated by the data.

Answer

\(\lim_{x\to4}H(x)\approx0.0020\). Writing \(0\) would erase the small but clearly nonzero limit suggested by the table.
54251012
A function \(K\) is defined for \(x>0\). The table samples values near the boundary \(x=0\). <table><tr><th>\(x\)</th><td>\(0.5\)</td><td>\(0.1\)</td><td>\(0.01\)</td><td>\(0.001\)</td></tr><tr><th>\(K(x)\)</th><td>\(1.581\)</td><td>\(1.449\)</td><td>\(1.418\)</td><td>\(1.415\)</td></tr></table> Estimate \(\lim_{x\to0^+}K(x)\) to three decimal places. Why is a right-hand limit the natural question here?

Hints

- Use the stated domain before interpreting the table. - Follow the outputs as the positive inputs approach \(0\). - Match the limit direction to the side of the domain that is available.

Solution

1. The function is defined only for positive inputs, so values can approach \(0\) only from the right within the domain. 2. As the positive inputs get closer to \(0\), the outputs approach about \(1.414\). 3. Therefore \(\lim_{x\to0^+}K(x)\approx1.414\).

Answer

\(\lim_{x\to0^+}K(x)\approx1.414\). A right-hand limit is natural because the domain contains nearby inputs only for \(x>0\).
54251112
A calculator displays the following rounded values of \(L(x)\) near \(x=8\). <table><tr><th>\(x\)</th><td>\(7.9\)</td><td>\(7.99\)</td><td>\(7.999\)</td><td>\(8.001\)</td><td>\(8.01\)</td><td>\(8.1\)</td></tr><tr><th>\(L(x)\)</th><td>\(12.0\)</td><td>\(12.0\)</td><td>\(12.0\)</td><td>\(12.0\)</td><td>\(12.0\)</td><td>\(12.0\)</td></tr></table> What limit estimate is supported by the displayed table? Explain why the rounded display does not prove that every nearby value of \(L(x)\) is exactly \(12\).

Hints

- Separate what the displayed values support from what they prove exactly. - Notice the stated rounding of the calculator output. - A numerical table can hide small differences that occur beyond the displayed digits.

Solution

1. All displayed values on both sides of \(8\) are \(12.0\), so the table supports \(\lim_{x\to8}L(x)\approx12\). 2. Each output has been rounded to one decimal place. 3. Different nearby values can round to the same display, so the table does not prove that \(L(x)=12\) exactly for every nearby input.

Answer

The table supports \(\lim_{x\to8}L(x)\approx12\). Because the outputs are rounded, identical displays do not imply the exact nearby function values are all \(12\).
52989712
A student is studying \(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\) and claims: “The expression inside the parentheses approaches \(1\), and \(1\) raised to an infinite power is \(1\), so the limit must be \(1\).” 1. Test the claim by calculating the expression for \(n=5\), \(n=50\), and \(n=500\). Round to three decimal places. 2. Explain why the student's reasoning is incomplete. Address how the base and exponent change together. 3. Name this limit and give its value rounded to two decimal places.

Hints

- Evaluate the expression for increasingly large values of \(n\). - What happens when a number slightly greater than \(1\) is multiplied by itself many times? - Are the computed values moving toward \(1\)? - Consider whether separate limiting behaviors can compete with each other.

Solution

1. For \(n=5\), \(\left(1+\frac{1}{5}\right)^5\approx2.488\). For \(n=50\), \(\left(1+\frac{1}{50}\right)^{50}\approx2.692\). For \(n=500\), \(\left(1+\frac{1}{500}\right)^{500}\approx2.716\). 2. The base approaches \(1\), but the exponent grows without bound at the same time. A number slightly greater than \(1\), multiplied by itself an increasing number of times, need not approach \(1\). The form \(1^{\infty}\) is indeterminate, so the combined limit must be analyzed rather than found by substituting the two separate limits. 3. The limit is Euler's number \(e\), and \(e\approx2.72\).

Answer

1. \(2.488\), \(2.692\), and \(2.716\) 2. The base and exponent change simultaneously; \(1^{\infty}\) is an indeterminate form. 3. Euler's number, \(e\approx2.72\)
54249612
Two functions are sampled near \(x=1\). <table><tr><th>\(x\)</th><td>\(0.9\)</td><td>\(0.99\)</td><td>\(0.999\)</td><td>\(1.001\)</td><td>\(1.01\)</td><td>\(1.1\)</td></tr><tr><th>\(a(x)\)</th><td>\(3.7\)</td><td>\(3.97\)</td><td>\(3.997\)</td><td>\(4.003\)</td><td>\(4.03\)</td><td>\(4.3\)</td></tr><tr><th>\(b(x)\)</th><td>\(3.8\)</td><td>\(3.98\)</td><td>\(3.998\)</td><td>\(5.002\)</td><td>\(5.02\)</td><td>\(5.2\)</td></tr></table> Estimate \(\lim_{x\to1}a(x)\) and determine whether \(\lim_{x\to1}b(x)\) exists. Use the table to explain why the conclusions differ.

Hints

- Treat each function row separately. - For each function, compare the trend from inputs less than \(1\) with the trend from inputs greater than \(1\). - A two-sided limit requires the two one-sided trends to agree.

Solution

1. For \(a\), values from the left approach \(4\), and values from the right also approach \(4\). Thus \(\lim_{x\to1}a(x)\approx4\). 2. For \(b\), left-side values approach \(4\), while right-side values approach \(5\). 3. Because the one-sided trends for \(b\) disagree, \(\lim_{x\to1}b(x)\) does not exist.

Answer

\(\lim_{x\to1}a(x)\approx4\). \(\lim_{x\to1}b(x)\) does not exist because the left-hand values approach \(4\) while the right-hand values approach \(5\).
54250212
A student wants to estimate \(\lim_{x\to3}p(x)\) from this table. <table><tr><th>\(x\)</th><td>\(2.9\)</td><td>\(2.99\)</td><td>\(2.999\)</td><td>\(3.001\)</td><td>\(3.01\)</td><td>\(3.1\)</td></tr><tr><th>\(p(x)\)</th><td>\(7.2\)</td><td>\(7.92\)</td><td>\(7.992\)</td><td>\(8.008\)</td><td>\(8.08\)</td><td>\(8.8\)</td></tr></table> The student averages the two farthest outputs, \(7.2\) and \(8.8\), and gets \(8\). The numerical answer happens to be reasonable. Explain why the student's method is not a reliable general method for estimating a limit, and give the estimate supported by the table.

Hints

- Ask which rows best represent behavior near \(x=3\). - A correct numerical result can come from an unjustified method. - Compare the closest outputs on opposite sides of the target.

Solution

1. A limit is determined by behavior for inputs close to the target, so the farthest rows are weaker evidence than the nearest rows. 2. The closest outputs, \(7.992\) and \(8.008\), approach \(8\) from opposite sides. 3. Thus the table supports \(\lim_{x\to3}p(x)\approx8\), but not because the farthest outputs happen to average to \(8\).

Answer

\(\lim_{x\to3}p(x)\approx8\). Averaging faraway outputs is not a reliable limit procedure; the estimate is supported by the values at inputs closest to \(3\).
55592912
The graph and table show the same function \(h\) near \(x=1\). The graph is displayed at a coarse vertical scale. <table><tr><th>\(x\)</th><td>\(0.9\)</td><td>\(0.99\)</td><td>\(0.999\)</td><td>\(1.001\)</td><td>\(1.01\)</td><td>\(1.1\)</td></tr><tr><th>\(h(x)\)</th><td>\(2.00100\)</td><td>\(2.00370\)</td><td>\(2.00397\)</td><td>\(1.99603\)</td><td>\(1.99630\)</td><td>\(1.99900\)</td></tr></table> a) Using only the graph and reporting to the nearest hundredth, estimate the left-hand and right-hand limits at \(x=1\). b) Using the table, estimate each one-sided limit to the nearest thousandth. c) Does \(\lim_{x\to1}h(x)\) exist? Explain why the coarse graphical estimates in part a) do not settle the question.
Figure for problem 555929

Hints

- Keep the precision requested in each part; the graph and table do not resolve values equally finely. - For the table, separate inputs less than \(1\) from inputs greater than \(1\) and focus on the closest entries. - A two-sided limit requires the actual one-sided limits to agree, not merely rounded estimates that look the same.

Solution

1. At the graph's displayed precision, both branches appear to approach about \(2.00\), so the nearest-hundredth graphical estimates agree. 2. The left-side values closest to \(1\) approach \(2.004\). 3. The right-side values closest to \(1\) approach \(1.996\). 4. Because \(2.004\ne1.996\), the two-sided limit does not exist. 5. The actual separation is only \(0.008\), so rounding graphical estimates to the nearest hundredth hides it. The table supplies the extra precision needed to distinguish the one-sided limits.

Answer

a) Left-hand limit \(\approx2.00\); right-hand limit \(\approx2.00\). b) Left-hand limit \(\approx2.004\); right-hand limit \(\approx1.996\). c) No. \(\lim_{x\to1}h(x)\) does not exist because the one-sided limits are unequal. The graph's nearest-hundredth precision rounds both to \(2.00\), masking the difference.

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Building free math worksheets