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AP Calculus AB : Asymptotic behavior in terms of limits involving infinity

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Example Questions

Example Question #11 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

$\begin{align*}&\text{Which of the four following functions is depicted in the sample data above?}\end{align*}$

Possible Answers:

$\frac{(6x - 5)}{(10x + 1)}$

$\frac{(10x + 1)}{(6x - 5)}$

$\frac{(10x - 1)}{(6x + 5)}$

$\frac{(6x + 5)}{(10x - 1)}$

Correct answer:

$\frac{(10x - 1)}{(6x + 5)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-0.8\\&\text{We find a zero denominator for the function: }\\&\frac{(10x - 1)}{(6x + 5)}\end{align*}$

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Example Question #11 : Asymptotic And Unbounded Behavior

Asymptoteplot

$\begin{align*}&\text{Of the following functions, which most likely represents the sample data above?}\end{align*}$

Possible Answers:

$\frac{(8x - 18)}{(16x - 12)}$

$\frac{(16x - 12)}{(8x - 18)}$

$-\frac{(16x - 12)}{(8x - 18)}$

$-\frac{(8x - 18)}{(16x - 12)}$

Correct answer:

$\frac{(16x - 12)}{(8x - 18)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=2\\&\text{This matches the ratio of coefficients for the function: }\\&\frac{(16x - 12)}{(8x - 18)}\end{align*}$

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Example Question #13 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

$\begin{align*}&\text{Decide which of the following functions, most likely represents the sample data above.}\end{align*}$

Possible Answers:

$\frac{(11x - 18)}{(15x - 17)}$

$\frac{(15x - 17)}{(11x - 18)}$

$-\frac{(11x - 18)}{(15x - 17)}$

$-\frac{(15x - 17)}{(11x - 18)}$

Correct answer:

$-\frac{(11x - 18)}{(15x - 17)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=-0.73\\&\text{This matches the ratio of coefficients for the function: }\\&-\frac{(11x - 18)}{(15x - 17)}\end{align*}$

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Example Question #11 : Functions, Graphs, And Limits

Asymptoteplot

$\begin{align*}&\text{Of the following functions, which most likely represents the sample data above?}\end{align*}$

Possible Answers:

$\frac{(17x + 4)}{(11x - 18)}$

$\frac{(11x + 18)}{(17x - 4)}$

$\frac{(11x - 18)}{(17x + 4)}$

$\frac{(17x - 4)}{(11x + 18)}$

Correct answer:

$\frac{(11x - 18)}{(17x + 4)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-0.2\\&\text{We find a zero denominator for the function: }\\&\frac{(11x - 18)}{(17x + 4)}\end{align*}$

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Example Question #11 : Asymptotic And Unbounded Behavior

Asymptoteplot

$\begin{align*}&\text{Decide which of the following functions, most likely represents the sample data above.}\end{align*}$

Possible Answers:

$\frac{(12x - 11)}{(18x + 17)}$

$\frac{(18x + 17)}{(12x - 11)}$

$-\frac{(18x + 17)}{(12x - 11)}$

$-\frac{(12x - 11)}{(18x + 17)}$

Correct answer:

$-\frac{(18x + 17)}{(12x - 11)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=-1.5\\&\text{This matches the ratio of coefficients for the function: }\\&-\frac{(18x + 17)}{(12x - 11)}\end{align*}$

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Example Question #11 : Asymptotic And Unbounded Behavior

Asymptoteplot

$\begin{align*}&\text{Which of the four following functions is most likely depicted in the sample data above?}\end{align*}$

Possible Answers:

$-\frac{(10x + 13)}{(x + 1)}$

$-\frac{(x - 1)}{(10x - 13)}$

$-\frac{(10x - 13)}{(x - 1)}$

$-\frac{(x + 1)}{(10x + 13)}$

Correct answer:

$-\frac{(x + 1)}{(10x + 13)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-1.3\\&\text{We find a zero denominator for the function: }\\&-\frac{(x + 1)}{(10x + 13)}\end{align*}$

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Example Question #17 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

$\begin{align*}&\text{Which of the four following functions is probably depicted in the sample data above?}\end{align*}$

Possible Answers:

$\frac{(5x - 15)}{(19x - 15)}$

$\frac{(19x - 15)}{(5x - 15)}$

$-\frac{(5x - 15)}{(19x - 15)}$

$-\frac{(19x - 15)}{(5x - 15)}$

Correct answer:

$-\frac{(19x - 15)}{(5x - 15)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=-3.8\\&\text{This matches the ratio of coefficients for the function: }\\&-\frac{(19x - 15)}{(5x - 15)}\end{align*}$

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Example Question #21 : Functions, Graphs, And Limits

Compute $\lim_{g\rightarrow \infty } \left ( \frac{g^4 -2g^5}{g^5 + g^3} \right )$

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

Firstly, recall the rules for evaluating the limits of rational expressions as they go to infinity:

If the degree on the top is greater than the degree on the bottom, the limit does not exist.

If the degree on the top is less than the degree on the bottom, the limit is 0.

If the degrees in both the numerator and denominator are equal, the limit is the ratio of the leading coefficients.

These rules follow from how the function grows as the inputs get larger, ignoring everything but the leading terms.

Then, note that, while the numerator is not written in the correct order for you, that the highest power in the numerator is 5. Likewise, the highest power in the denominator is 5. Thus, the limit will be the ratio of the coefficients on the g^5 terms, which is $\frac{-2}{1} = -2$ .

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Example Question #21 : Functions, Graphs, And Limits

Evaluate

$\lim_{x\rightarrow \infty}\frac{4x^3+2x^2-5x+1}{x^3-x^2+3x-5}$

Possible Answers:

$\infty$

$\frac{-1}{5}$

Correct answer:

Explanation:

The equation $\frac{4x^3+2x^2-5x+1}{x^3-x^2+3x-5}$ will have a horizontal asymptote y=4.

We can find the horizontal asymptote by looking at the terms with the highest power.

The terms with the highest power here are 4x^3 in the numerator and x^3 in the denominator. These terms will "take over" the function as x approaches infinity. That means the limit will reach the ratio of the two terms.

The ratio is $\frac{4x^3}{x^3}=\frac{4}{1}=4$

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Example Question #21 : Functions, Graphs, And Limits

Find $\lim_{x\rightarrow \infty}\frac{ln(x)}{x}$ .

Possible Answers:

$\infty$

Correct answer:

Explanation:

First step for finding limits: evaluate the function at the limit.

$\frac{ln(\infty )}{\infty }=\frac{\infty }{\infty }$ which is an Indeterminate Form.

This got us nowhere. However, since we have an indeterminate form, we can use L'Hopital's Rule (take the derivative of the top and bottom and the limit's value won't change).

$\lim_{x\rightarrow \infty}\frac{ln(x)}{x}=\lim_{x\rightarrow \infty }\frac{1/x}{1}=\lim_{x\rightarrow \infty}\frac{1}{x}$

This is something we can evaluate:

$\lim_{x\to\infty}\frac{1}{x}\rightarrow \frac{1}{\infty}\rightarrow 0$

The value of this limit is .

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AP Calculus AB : Asymptotic behavior in terms of limits involving infinity

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #11 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Example Question #11 : Asymptotic And Unbounded Behavior

Example Question #13 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Example Question #11 : Functions, Graphs, And Limits

Example Question #11 : Asymptotic And Unbounded Behavior

Example Question #11 : Asymptotic And Unbounded Behavior

Example Question #17 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Example Question #21 : Functions, Graphs, And Limits

Example Question #21 : Functions, Graphs, And Limits

Example Question #21 : Functions, Graphs, And Limits

All AP Calculus AB Resources

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