Free math worksheets

- Locate \(x=2.5\) halfway between the labeled integer inputs. - Read the graph height using the quarter-unit horizontal grid. - Check that the same height is approached from both sides.

1. Trace the graph from both sides toward \(x=2.5\). 2. The y-values approach approximately \(2.75\) from both sides. 3. Therefore \(\lim_{x\to2.5}p(x)\approx2.75\).

\(\lim_{x\to2.5}p(x)\approx2.75\).

- Follow the curve toward \(x=1\) from both sides. - Use the y-value approached by the curve, not automatically the filled point. - Check whether the left and right sides approach the same height.

1. From both sides of \(x=1\), the graph approaches the open point \((1, 0)\). 2. Thus the left-hand and right-hand limits both equal \(0\). 3. Therefore \(\lim_{x\to1}f(x)=0\). 4. The filled point at \((1, 3)\) gives \(f(1)=3\) but does not change the nearby limiting behavior.

\(\lim_{x\to1}f(x)=0\). The filled point gives \(f(1)=3\), which does not determine the limit.

- Trace the graph toward \(x=-1\) separately from the left and from the right. - Read the height approached on each side. - A two-sided limit exists only when the two one-sided values agree.

1. From the left, the graph approaches the open point at \((-1, 2)\), so \(\lim_{x\to-1^-}g(x)=2\). 2. From the right, the graph approaches the open point at \((-1, -1)\), so \(\lim_{x\to-1^+}g(x)=-1\). 3. Because the one-sided limits are unequal, \(\lim_{x\to-1}g(x)\) does not exist.

\(\lim_{x\to-1^-}g(x)=2\), \(\lim_{x\to-1^+}g(x)=-1\), and \(\lim_{x\to-1}g(x)\) does not exist.

- A limit depends on the y-value approached, not on whether the graph is smooth. - Trace both arms of the graph toward \(x=1\). - Compare the two one-sided heights.

1. From the left, the graph approaches the point \((1, 2)\). 2. From the right, the graph also approaches \((1, 2)\). 3. Thus \(\lim_{x\to1}g(x)=2\). 4. A corner does not prevent a limit when both sides approach the same y-value.

\(\lim_{x\to1}g(x)=2\). The corner does not prevent the limit from existing.

- Watch what happens to the height of each branch near \(x=0\). - Ask whether the branches settle near any finite y-value. - A graph leaving every finite vertical scale indicates unbounded behavior rather than a finite limit.

1. On both sides of \(x=0\), the graph rises beyond the visible y-range as \(x\) moves closer to \(0\). 2. The y-values do not approach any finite height. 3. Therefore \(\lim_{x\to0}r(x)\) does not exist as a finite real number; the behavior is unbounded above.

No finite real limit exists. The graph is unbounded above as \(x\to0\) from both sides.

- Locate \(x=1.5\) using the half-unit grid. - Follow the curve toward that vertical line from both sides. - Read the corresponding y-value using the same grid spacing.

1. The curve is continuous near \(x=1.5\). 2. Reading the graph at \(x=1.5\), the y-value approached from both sides is about \(1.0\). 3. Therefore \(\lim_{x\to1.5}p(x)\approx1.0\).

\(\lim_{x\to1.5}p(x)\approx 1.0\).

- Read each branch separately as it moves toward the y-axis. - Distinguish decreasing without bound from increasing without bound. - Compare the two one-sided behaviors before stating a two-sided conclusion.

1. From the left, the graph decreases without bound as it approaches \(x=0\), so \(f(x)\to-\infty\). 2. From the right, the graph increases without bound as it approaches \(x=0\), so \(f(x)\to+\infty\). 3. The one-sided behaviors do not approach the same finite value, so no finite two-sided limit exists.

\(\lim_{x\to0^-}f(x)=-\infty\), \(\lim_{x\to0^+}f(x)=+\infty\), and no finite two-sided limit exists.

- Identify the two endpoints of the displayed domain. - Approach each endpoint from the side where the graph exists. - Do not infer real-valued behavior where the function has no real inputs.

1. As \(x\to-3^+\), the graph approaches y-value \(0\), so the right-hand limit is \(0\). 2. As \(x\to3^-\), the graph also approaches y-value \(0\), so the left-hand limit is \(0\). 3. The graph has no real-valued inputs below \(-3\) or above \(3\), so the opposite one-sided approaches are unavailable in the real domain.

\(\lim_{x\to-3^+}q(x)=0\) and \(\lim_{x\to3^-}q(x)=0\). The opposite sides are outside the real domain.

- Follow the graph toward its left endpoint. - Read the y-value approached as \(x\) gets close to \(0\) from within the domain. - Check whether the real domain contains inputs to the left of \(0\).

1. As \(x\) approaches \(0\) through positive values, the graph approaches \((0, 0)\). 2. Therefore \(\lim_{x\to0^+}h(x)=0\). 3. The real-valued function \(\sqrt{x}\) has no domain points with \(x<0\), so there is no real graph to approach \(0\) from the left.

\(\lim_{x\to0^+}h(x)=0\). There is no real-valued left-hand approach because \(\sqrt{x}\) is not defined for \(x<0\).

- Trace the graph toward each input from the left and record the y-value approached. - Repeat from the right. - A two-sided limit exists only when the two one-sided limits agree. - The function value at the input does not determine the limit by itself.

1. As \(x\) approaches \(1\) from the left, \(f(x)\) approaches \(2\). As \(x\) approaches \(1\) from the right, \(f(x)\) also approaches \(2\). Therefore, \(\lim_{x\to1}f(x)=2\). The separate value \(f(1)=1\) does not affect the existence or value of the limit. 2. As \(x\) approaches \(3\) from the left, \(f(x)\) approaches \(3\). As \(x\) approaches \(3\) from the right, \(f(x)\) approaches \(1\). Since the one-sided limits are different, \(\lim_{x\to3}f(x)\) does not exist.

At \(x=1\): \(\lim_{x\to1}f(x)=2\). At \(x=3\): \(\lim_{x\to3}f(x)\) does not exist because the left-hand limit is \(3\) and the right-hand limit is \(1\).

- Look at the vertical spread of the graph very close to \(x=0\). - Ask whether the y-values settle into a narrow band around one number. - A finite limit requires the nearby y-values to fit inside increasingly narrow bands around one number.

1. Near \(x=0\), the graph repeatedly takes values near \(1\) and \(-1\) instead of settling near one height. 2. Thus the function values do not approach a single finite number from either side. 3. Therefore \(\lim_{x\to0}p(x)\) does not exist.

No. \(\lim_{x\to0}p(x)\) does not exist because the graph continues to oscillate rather than approaching one value.

- Analyze the two sides separately before combining them. - On the right, ask whether the branch stays near any finite height. - A finite two-sided limit requires the same finite value from both sides.

1. The left branch is horizontal at y-value \(2\), so \(\lim_{x\to0^-}r(x)=2\). 2. The right branch rises without bound as \(x\to0^+\). 3. Since the two sides do not approach the same finite value, no finite two-sided limit exists.

a) \(2\) b) \(r(x)\) increases without bound as \(x\to0^+\). c) No finite two-sided limit exists.

- Analyze each panel independently. - For the first two panels, compare the heights approached from the left and right. - For the oscillating panel, ask whether the y-values settle near one number as \(x\) gets closer to \(0\).

1. In panel a), both branches approach y-value \(2\), so \(\lim_{x\to0}f(x)=2\). 2. In panel b), the left branch approaches \(-1\) and the right branch approaches \(3\), so the two-sided limit does not exist. 3. In panel c), the graph keeps oscillating between separated y-values near \(0\), so it does not approach a single value and the limit does not exist.

a) \(\lim_{x\to0}f(x)=2\). b) \(\lim_{x\to0}g(x)\) does not exist. c) \(\lim_{x\to0}h(x)\) does not exist.

- Treat each x-value as a separate local question. - At \(-2\), distinguish the open point from the filled point. - At \(0\), compare the two branch heights. - At \(2\), follow the same continuous branch from both sides.

1. At \(x=-2\), both sides of the first branch approach y-value \(1\), so \(\lim_{x\to-2}F(x)=1\), even though the plotted function value is different. 2. At \(x=0\), the left branch approaches \(2\) while the right branch approaches \(-2\), so the two-sided limit does not exist. 3. At \(x=2\), the right-side branch is continuous through y-value \(-1\), and nearby values from both sides of \(2\) on that branch approach \(-1\). 4. Thus \(\lim_{x\to2}F(x)=-1\).

a) \(1\) b) \(\lim_{x\to0}F(x)\) does not exist. c) \(-1\)

- Oscillation alone does not determine whether a limit exists. - Track the vertical size of the oscillations near \(x=0\). - Ask whether all nearby y-values fit inside increasingly narrow bands around one number.

1. The graph alternates above and below the x-axis near \(x=0\). 2. The vertical distance of those oscillations from \(0\) becomes smaller as the inputs approach \(0\). 3. Both sides are confined closer and closer to y-value \(0\), so \(\lim_{x\to0}g(x)=0\).

\(\lim_{x\to0}g(x)=0\). The oscillations shrink toward y-value \(0\).

- Inspect the two branches next to \(x=-1\) separately. - Use the direction of unbounded motion to choose the sign of infinity. - For the end behavior, look for the y-value the graph approaches as \(x\) moves right.

1. The left branch decreases without bound as \(x\to-1^-\), so the left-hand limit is \(-\infty\). 2. The right branch increases without bound as \(x\to-1^+\), so the right-hand limit is \(+\infty\). 3. Far to the right, the graph levels off near y-value \(1\), so \(\lim_{x\to\infty}h(x)=1\).

\(\lim_{x\to-1^-}h(x)=-\infty\), \(\lim_{x\to-1^+}h(x)=+\infty\), and \(\lim_{x\to\infty}h(x)=1\).

- First read the limit of the original graph near \(x=0\). - Then apply the absolute value to the y-value being approached. - Separate the nearby behavior from the isolated filled point.

1. Both graph branches approach y-value \(-2\) as \(x\to0\), so \(f(x)\to-2\). 2. Taking absolute values makes the nearby outputs approach \(|-2|=2\). 3. The filled point gives \(f(0)=3\), but a single point value does not change the limit.

\(\lim_{x\to0}|f(x)|=2\). The filled point does not affect the limit.

- Translate the output condition into a horizontal band on the graph. - For each candidate, inspect the full punctured interval around \(x=1\). - Choose the largest listed interval that remains inside the required band.

1. The condition \(|r(x)-2|<1\) means the graph must stay between y-values \(1\) and \(3\). 2. Within \(1\) unit of \(x=1\), the displayed graph stays inside that horizontal band. 3. At inputs about \(1.5\) units from \(1\), the graph rises above \(3\), so \(\delta=1.5\) is not supported. 4. The largest listed value supported by the graph is \(\delta=1\).

\(\delta=1\)

- For each input, compare the y-values approached from the left and right. - An isolated filled point does not determine a limit. - Unbounded behavior is not a finite limit.

1. At \(x=-2\), both branches approach y-value \(1\), so the finite two-sided limit is \(1\). 2. At \(x=1\), the left branch approaches \(2\) and the right branch approaches \(-1\), so the two-sided limit does not exist. 3. At \(x=3\), the graph is unbounded on the two sides, so no finite two-sided limit exists. 4. Therefore only \(x=-2\) has a finite two-sided limit among the listed inputs.

At \(x=-2\), the finite two-sided limit exists and equals \(1\). At \(x=1\), the two-sided limit does not exist. At \(x=3\), no finite two-sided limit exists.