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AP Calculus BC : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #72 : Limits

For the piecewise function:

$y=\left\{\begin{matrix} 1 \textup{, for }x> 0\\ 0 \textup{, for }x< 0\end{matrix}\right.$ , find $\lim_{x\to 0^+}$ .

Possible Answers:

Any real number.

Does not exist.

$\frac{1}{2}$

Correct answer:

Explanation:

The limit $\lim_{x\to 0^+}$ indicates that we are trying to find the value of the limit as approaches to zero from the right side of the graph.

From right to left approaching x=0 , the limit approaches to 1 even though the value at x=0 of the piecewise function does not exist.

The answer is .

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Example Question #74 : Limits

Screen shot 2015 07 07 at 1.28.45 pm

Given the graph of f(x) above, what is $\lim_{x\rightarrow 0}f(x)$ ?

Possible Answers:

$\infty$

$-\infty$

Correct answer:

Explanation:

Examining the graph of the function above, we need to look at three things:

1) What is the limit of the function as it approaches zero from the left?

2) What is the limit of the function as it approaches zero from the right?

3) What is the function value at zero and is it equal to the first two statements?

If we look at the graph we see that as approaches zero from the left the values approach zero as well. This is also true if we look the values as approaches zero from the right. Lastly we look at the function value at zero which in this case is also zero.

Therefore, we can observe that $\lim_{x\rightarrow 0}f(x)=0$ as approaches .

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Example Question #77 : Limits

Screen shot 2015 07 07 at 4.36.36 pm

Given the graph of f(x) above, what is $\lim_{x\rightarrow 0}f(x)$ ?

Possible Answers:

Does not exist

$\infty$

$-\infty$

Correct answer:

Does not exist

Explanation:

Examining the graph above, we need to look at three things:

1) What is the limit of the function as approaches zero from the left?

2) What is the limit of the function as approaches zero from the right?

3) What is the function value as x=0 and is it the same as the result from statement one and two?

Therefore, we can determine that $\lim_{x\rightarrow 0}f(x)$ does not exist, since approaches two different limits from either side : $-\infty$ from the left and $\infty$ from the right.

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Example Question #82 : Limits

Limit b 7.15

Given the above graph of f(x) , what is $\lim_{x\rightarrow 0^{+}}f(x)$ ?

Possible Answers:

$\infty$

$-\infty$

Correct answer:

$\infty$

Explanation:

Examining the graph, we want to find where the graph tends to as it approaches zero from the right hand side. We can see that there appears to be a vertical asymptote at zero. As the x values approach zero from the right the function values of the graph tend towards positive infinity.

Therefore, we can observe that $\lim_{x\rightarrow 0^{+}}f(x)=\infty$ as approaches from the right.

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Example Question #41 : Functions, Graphs, And Limits

Screen shot 2015 08 17 at 11.29.05 am

Given the above graph of f(x) , what is $\lim_{x\rightarrow 0}f(x)$ ?

Possible Answers:

Does Not Exist

$-\infty$

$\infty$

Correct answer:

Does Not Exist

Explanation:

Examining the graph, we can observe that $\lim_{x\rightarrow 0}f(x)$ does not exist, as f(x) is not continuous at x=0 . We can see this by checking the three conditions for which a function f(x) is continuous at a point (c, f(c)) :

A value exists in the domain of
The limit of exists as approaches
The limit of at is equal to

Given c=0 , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for f(0) and is therefore an infinite discontinuity at (0,f(0)) .

We can also see that condition #2 is not satisfied because $\lim_{x\rightarrow 0}f(x)$ approaches two different limits: $\infty$ from the left and $-\infty$ from the right.

Based on the above, condition #3 is also not satisfied because $\lim_{x\rightarrow 0}f(x)$ is not equal to the multiple values of f(c) .

Thus, $\lim_{x\rightarrow 0}f(x)$ does not exist.

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Example Question #1 : Limits

Limitplot

$\begin{align*}&\text{What is the likely value of}\\&f(-\infty)\\&\text{Based on the graph?}\end{align*}$

Possible Answers:

$\text{Nonexistent.}$

$\infty$

$-\infty$

Correct answer:

$\text{Nonexistent.}$

Explanation:

$\begin{align*}&\text{Although actually observing a function at an infinite value}\\&\text{of x is impossible, we can often determine via observation if}\\&\text{the function approaches a certain limit. Looking at the values}\\&\text{the function approaches as the absolute value of x increases}\\&\text{gives a clue. Things to look out for are if the function continuously}\\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\\&\text{If the function continuously climbs/decreases, it likely has}\\&\text{a limit that approaches positive/negative infinity. If it flattens,}\\&\text{the limit is a real number. If it oscillates without control,}\\&\text{the limit does not exist.}\\&\text{The limit of }f(-\infty)\text{ is most likely }\text{Nonexistent.}\end{align*}$

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Example Question #1 : Estimating Limits From Graphs And Tables

Limitplot

$\begin{align*}&\text{What is the likely value of}\\&f(-\infty)\\&\text{Based on the graph?}\end{align*}$

Possible Answers:

$-\infty$

$\frac{3}{2}$

$\text{Nonexistent.}$

Correct answer:

Explanation:

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Example Question #5 : Limits

Limitplot

$\begin{align*}&\text{Which of the values most likely represents the value of f(x) at}\\&x=\infty\end{align*}$

Possible Answers:

$\text{Nonexistent.}$

$\infty$

Correct answer:

Explanation:

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Example Question #1 : Limits

Limitplot

$\begin{align*}&\text{Based on the graph, decide which of the following functions values likely depicts}\\&f(-\infty)\end{align*}$

Possible Answers:

$-\infty$

$\text{Nonexistent.}$

$-\frac{5}{7}$

Correct answer:

$\text{Nonexistent.}$

Explanation:

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Example Question #1 : Estimating Limits From Graphs And Tables

Limitplot

$\begin{align*}&\text{What is the likely value of}\\&f(-\infty)\\&\text{Based on the graph?}\end{align*}$

Possible Answers:

$-\frac{2}{3}$

$\text{Nonexistent.}$

$\infty$

Correct answer:

$\infty$

Explanation:

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AP Calculus BC : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #72 : Limits

Example Question #74 : Limits

Example Question #77 : Limits

Example Question #82 : Limits

Example Question #41 : Functions, Graphs, And Limits

Example Question #1 : Limits

Example Question #1 : Estimating Limits From Graphs And Tables

Example Question #5 : Limits

Example Question #1 : Limits

Example Question #1 : Estimating Limits From Graphs And Tables

All AP Calculus BC Resources

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