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Limit notation

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54246612
Write the statement “As \(x\) approaches \(3\), the values of \(f(x)\) approach \(-2\)” using standard limit notation.

Hints

- Identify the input variable and the number it approaches. - Identify the function whose output is being described. - Put the approached output on the right side of the limit statement.

Solution

1. The input variable is \(x\), the approach value is \(3\), and the function values approach \(-2\). 2. Standard notation places these pieces in \(\lim_{x\to3}f(x)=-2\).

Answer

\(\lim_{x\to3}f(x)=-2\).
54246712
Interpret the statement \(\lim_{t\to5^-}P(t)=12\) in words. Your interpretation should make clear what the superscript minus sign means.

Hints

- Focus on the approach symbol under the limit. - Decide whether the superscript describes the sign of the function value or the side of the input. - Separate the limiting behavior from the actual value at \(t=5\).

Solution

1. The notation \(t\to5^-\) means \(t\) approaches \(5\) through values less than \(5\). 2. Along that approach, the values \(P(t)\) approach \(12\). 3. The statement does not by itself specify \(P(5)\) or behavior from the right.

Answer

As \(t\) approaches \(5\) from values less than \(5\), \(P(t)\) approaches \(12\).
54247112
A student reads \(\lim_{x\to2}f(x)=5\) as “the input \(x\) approaches \(5\) while the output approaches \(2\).” Correct the student's interpretation.

Hints

- Look at the subscript to identify the input behavior. - Look at the right side of the equation to identify the approached output. - Do not reverse the roles of \(2\) and \(5\).

Solution

1. The expression under the limit is \(f(x)\), so the limit concerns the outputs of \(f\). 2. The subscript \(x\to2\) says the input approaches \(2\). 3. The value on the right side, \(5\), is the output value approached by \(f(x)\).

Answer

As \(x\) approaches \(2\), the values of \(f(x)\) approach \(5\).
54247612
A student writes \(\lim_{u\to0}H(x)=4\) while discussing how \(H(u)\) behaves as \(u\) approaches \(0\). Rewrite the statement with consistent notation and explain the correction.

Hints

- Compare the variable under the limit with the variable inside the function. - Keep the input symbol consistent throughout the statement. - Check that the approached input and output values remain in their correct positions.

Solution

1. The variable in the function expression should match the variable named in the approach instruction. 2. The consistent statement is \(\lim_{u\to0}H(u)=4\). 3. This states that the outputs \(H(u)\) approach \(4\) as the input \(u\) approaches \(0\).

Answer

\(\lim_{u\to0}H(u)=4\).
54247912
A student writes \(\lim_{3\to x}f(x)=8\) to mean that \(f(x)\) approaches \(8\) as \(x\) approaches \(3\). Rewrite the notation correctly and explain the order of the symbols in the approach expression.

Hints

- Ask which symbol is varying. - Ask which number is fixed as the target of the approach. - Keep the function expression unchanged while correcting the subscript.

Solution

1. The variable that changes must appear before the arrow. 2. The fixed approach value appears after the arrow. 3. The correct notation is \(\lim_{x\to3}f(x)=8\).

Answer

\(\lim_{x\to3}f(x)=8\). The variable goes before the arrow and the value it approaches goes after the arrow.
52190312
Let \(f(x)=\begin{cases}\frac{2x+8}{x+5}, &x<-3\\x^2+5x+c, &x>-3\end{cases}\), where \(c\) is real. a) Find \(\lim_{x\to-3^-}f(x)\). b) Find the value of \(c\) for which \(\lim_{x\to-3}f(x)\) exists.

Hints

- Use the formula that applies on the side from which \(x\) approaches \(-3\). - A two-sided limit exists only when the left- and right-hand limits are equal.

Solution

1. For the left-hand limit, use the first formula: \(\lim_{x\to-3^-}\frac{2x+8}{x+5}=\frac{2}{2}=1\). 2. For the right-hand limit, use the second formula: \(\lim_{x\to-3^+}(x^2+5x+c)=9-15+c=c-6\). 3. The two-sided limit exists when the one-sided limits are equal: \(1=c-6\). Thus, \(c=7\).

Answer

a) \(1\) b) \(c=7\)
52614012
A hypothetical savings account earns \(100\%\) annual interest. If the year is divided into \(n\) equal compounding periods, an initial balance of \(\$100.00\) grows after one year to \(K_n=100\left(1+\frac{1}{n}\right)^n\) dollars. 1. Find the ending balance for annual compounding \((n=1)\), monthly compounding \((n=12)\), and daily compounding \((n=365)\). Round to the nearest cent. 2. Find \(\lim_{n\to\infty}K_n\), representing continuous compounding. 3. Find the difference between the daily-compounding balance and the continuous-compounding balance.

Hints

- Substitute each value of \(n\) into the model. - Recall the limit that defines \(e\). - Keep extra decimal places until the final subtraction.

Solution

1. \(K_1=100(2)=\$200.00\). 2. \(K_{12}=100\left(1+\frac{1}{12}\right)^{12}\approx\$261.30\). 3. \(K_{365}=100\left(1+\frac{1}{365}\right)^{365}\approx\$271.46\). 4. Since \(\left(1+\frac{1}{n}\right)^n\to e\), \(K_n\to100e\approx\$271.83\). 5. The difference is \(100e-K_{365}\approx\$0.37\).

Answer

1. \(\$200.00\), \(\$261.30\), and \(\$271.46\) 2. \(100e\approx\$271.83\) 3. Approximately \(\$0.37\)
54246812
Suppose \(f(4)=9\) and \(\lim_{x\to4}f(x)=7\). A student says these two facts contradict each other. Explain why there is no contradiction, using the meaning of limit notation.

Hints

- Separate what happens near \(x=4\) from what happens exactly at \(x=4\). - Recall that a limit statement does not require the input to equal the approach value. - Ask which statement concerns one point and which concerns nearby behavior.

Solution

1. The statement \(\lim_{x\to4}f(x)=7\) describes \(f(x)\) for inputs \(x\) close to \(4\) but not necessarily equal to \(4\). 2. The equation \(f(4)=9\) describes only the function value at the single input \(4\). 3. A function can therefore have limit \(7\) at \(4\) while its assigned value at \(4\) is \(9\).

Answer

There is no contradiction. The limit describes nearby values of \(f(x)\), while \(f(4)=9\) describes the value at the point itself.
54246912
A student wants to express “\(g(x)\) approaches \(6\) as \(x\) approaches \(-1\)” and writes \(\lim g(-1)=6\). Identify what is missing or incorrect, and write the statement in standard limit notation.

Hints

- A limit needs an approach instruction for the input. - Keep the function written with a variable while describing nearby behavior. - Distinguish evaluation at a point from taking a limit toward that point.

Solution

1. Limit notation must show the variable and the value that variable approaches. 2. Writing \(g(-1)\) substitutes the point into the function instead of describing nearby inputs. 3. The correct statement is \(\lim_{x\to-1}g(x)=6\).

Answer

The correct notation is \(\lim_{x\to-1}g(x)=6\).
54247012
You are told that \(\lim_{x\to2^-}h(x)=4\) and \(\lim_{x\to2^+}h(x)=4\). Write the corresponding two-sided limit statement. Then state what additional information, if any, is needed about \(h(2)\) to write that limit statement.

Hints

- Compare the two one-sided limiting values. - Recall the condition for a finite two-sided limit to exist. - Keep the function value at the point separate from the limiting behavior.

Solution

1. The left-hand and right-hand limits both exist and equal \(4\). 2. Therefore the two-sided limit exists and equals \(4\). 3. The value \(h(2)\) is not needed to state the two-sided limit. 4. The limit statement is \(\lim_{x\to2}h(x)=4\).

Answer

\(\lim_{x\to2}h(x)=4\). No information about \(h(2)\) is needed for this limit statement.
54247212
A function \(r\) is not defined at \(x=-3\), but its values approach \(8\) as \(x\) approaches \(-3\) from either side. Write a limit statement that represents this information. Then explain why the missing function value does not prevent you from writing it.

Hints

- Separate the value of the function at the point from the values near the point. - Put the approach value with the input variable under the limit. - Put the approached output on the right side.

Solution

1. The nearby inputs approach \(-3\), and the nearby outputs approach \(8\). 2. The correct statement is \(\lim_{x\to-3}r(x)=8\). 3. A limit describes behavior for inputs arbitrarily close to the point, so \(r(-3)\) need not be defined.

Answer

\(\lim_{x\to-3}r(x)=8\). The limit can exist even though \(r(-3)\) is undefined.
54247312
Suppose \(\lim_{x\to0^-}p(x)=1\) and \(\lim_{x\to0^+}p(x)=5\). a) State each one-sided behavior in words. b) Does \(\lim_{x\to0}p(x)\) exist? Explain using the notation given.

Hints

- Read the superscript minus and plus as directions of approach. - Compare the two approached output values. - A two-sided limit requires the left- and right-hand limits to agree.

Solution

1. From the left, as \(x\) approaches \(0\), \(p(x)\) approaches \(1\). 2. From the right, as \(x\) approaches \(0\), \(p(x)\) approaches \(5\). 3. Because the one-sided limits are different, the two-sided limit does not exist.

Answer

a) From the left, \(p(x)\to1\); from the right, \(p(x)\to5\) as \(x\to0\). b) \(\lim_{x\to0}p(x)\) does not exist because the one-sided limits are unequal.
54247412
A student sees the notation \(\lim_{x\to4^-}f(x)=9\) and says, “The function values approach \(9\) from below.” Is that interpretation required by the notation? Explain what the superscript minus actually tells you.

Hints

- Locate exactly where the superscript minus appears. - Distinguish the direction of the input from the direction of the output. - A limit value alone does not describe whether outputs stay above or below that value.

Solution

1. The superscript minus is attached to the input approach \(x\to4^-\). 2. It says \(x\) approaches \(4\) through values less than \(4\). 3. The notation does not specify whether \(f(x)\) approaches \(9\) from above, from below, or by alternating around \(9\).

Answer

No. The superscript minus describes the input approaching \(4\) from the left; it does not specify the side from which \(f(x)\) approaches \(9\).
54247512
Assume \(\lim_{x\to c}f(x)=L\). Which statement is guaranteed? A. \(f(c)=L\) B. \(f(c)\) is defined C. The values \(f(x)\) can be made close to \(L\) by taking \(x\) sufficiently close to \(c\), with \(x\ne c\) D. \(f(x)=L\) for every \(x\) near \(c\) Choose the best answer and explain.

Hints

- Focus on behavior near the approach point rather than exactly at it. - A limit concerns closeness, not exact equality at every nearby input. - Eliminate claims that require a particular value of \(f(c)\).

Solution

1. A finite limit describes the behavior of \(f(x)\) for inputs near \(c\), not necessarily at \(c\). 2. It does not require \(f(c)\) to exist or equal \(L\), and it does not require nearby outputs to equal \(L\) exactly. 3. Choice C correctly states the meaning of the limit.

Answer

C.
54247712
The function \(r(x)=\sqrt{x}\) is defined only for \(x\ge0\). Write limit notation for the statement “As \(x\) approaches \(0\) through values in the domain, \(r(x)\) approaches \(0\).”

Hints

- Check which side of \(0\) belongs to the function's real domain. - Use one-sided notation when only one side is available. - Put the approached output after the equals sign.

Solution

1. Near \(0\), the domain contains inputs only to the right of \(0\). 2. Therefore the appropriate approach is \(x\to0^+\). 3. The statement is \(\lim_{x\to0^+}\sqrt{x}=0\).

Answer

\(\lim_{x\to0^+}\sqrt{x}=0\).
54247812
Interpret \(\lim_{h\to0}A(3+h)=7\). Describe what happens to the input of \(A\) as \(h\) approaches \(0\), and what happens to the output.

Hints

- Track the entire input expression \(3+h\), not just \(h\). - Ask what value \(3+h\) gets close to. - Then identify the approached output from the right side of the limit equation.

Solution

1. As \(h\to0\), the input \(3+h\) approaches \(3\). 2. The values \(A(3+h)\) approach \(7\). 3. Thus the statement describes the behavior of \(A\) near input \(3\), expressed using a shifted variable.

Answer

As \(h\) approaches \(0\), the input \(3+h\) approaches \(3\), and \(A(3+h)\) approaches \(7\).
54248012
A student says that \(x\to-3^-\) means “\(x\) is negative and approaches \(-3\).” Improve the statement so that it captures the role of both minus signs accurately.

Hints

- Separate the sign of the number \(-3\) from the one-sided superscript. - Translate “from the left” into an inequality involving nearby inputs. - Check which values lie to the left of \(-3\).

Solution

1. The minus sign in \(-3\) is part of the approach value itself. 2. The superscript minus indicates that \(x\) approaches \(-3\) from the left, through values less than \(-3\). 3. Thus the two minus signs serve different purposes.

Answer

\(x\to-3^-\) means that \(x\) approaches the negative number \(-3\) through values less than \(-3\).
52190412
Let \(g(x)=\begin{cases}4-x^2, &x<-1\\\frac{6}{x-1}+6, &-1<x<2\\2x+a, &x>2\end{cases}\), where \(a\) is real. a) Show that \(\lim_{x\to-1}g(x)\) exists, and find it. b) Analyze \(\lim_{x\to2}g(x)\) in terms of \(a\). For which value of \(a\) does the limit exist?

Hints

- Evaluate the one-sided limits using the corresponding formulas. - Set the left- and right-hand limits equal to determine the parameter.

Solution

1. At \(x=-1\), the left-hand limit is \(4-(-1)^2=3\). 2. The right-hand limit is \(\frac{6}{-1-1}+6=-3+6=3\). Since the one-sided limits agree, the limit exists and equals \(3\). 3. At \(x=2\), the left-hand limit is \(\frac{6}{2-1}+6=12\). 4. The right-hand limit is \(2(2)+a=4+a\). The two-sided limit exists when \(12=4+a\), so \(a=8\). For any other value of \(a\), the one-sided limits differ.

Answer

a) The limit exists and equals \(3\). b) The left-hand limit is \(12\), and the right-hand limit is \(4+a\). The limit exists only when \(a=8\), in which case it equals \(12\).

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