Asymptotic behavior in terms of limits involving infinity

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

Questions 1 - 10

Asymptoteplot

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

$-\frac{(x - 11)}{(4x + 1)}$

$-\frac{(4x + 1)}{(x - 11)}$

$-\frac{(x + 11)}{(4x - 1)}$

$-\frac{(4x - 1)}{(x + 11)}$

Explanation

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\&\text{and decrease on either side of a particular x-value. This type}\&\text{of behavior is observed when there’s a vertical asymptote. An}\&\text{asymptote is a value that a function may approach, but will}\&\text{never actually attain. In the case of vertical asymptotes, this}\&\text{behavior occurs if the function approaches infinity for a given}\&\text{x-value, often when a zero value appears in a denominator. Noting}\&\text{this rule, the above function has a zero in the denominator}\&\text{at a definite point centered approximately around:}\&x=-0.3\&\text{We find a zero denominator for the function: }\&-\frac{(x - 11)}{(4x + 1)}\end{align*}$

Asymptoteplot

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

$-\frac{(10x + 11)}{(11x + 20)}$

$-\frac{(11x + 20)}{(10x + 11)}$

$-\frac{(10x - 11)}{(11x - 20)}$

$-\frac{(11x - 20)}{(10x - 11)}$

Explanation

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\&\text{and decrease on either side of a particular x-value. This type}\&\text{of behavior is observed when there’s a vertical asymptote. An}\&\text{asymptote is a value that a function may approach, but will}\&\text{never actually attain. In the case of vertical asymptotes, this}\&\text{behavior occurs if the function approaches infinity for a given}\&\text{x-value, often when a zero value appears in a denominator. Noting}\&\text{this rule, the above function has a zero in the denominator}\&\text{at a definite point centered approximately around:}\&x=-1.8\&\text{We find a zero denominator for the function: }\&-\frac{(10x + 11)}{(11x + 20)}\end{align*}$

Asymptoteplot

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

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

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

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

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

Explanation

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

Assume that a population of bunnies grows at a rate of $\frac{\mathrm{d} P}{\mathrm{d} t} = 5P\left ( \frac{350 - P}{350} \right )$ where is the number of bunnies in the population at time . If the population begins with 14 bunnies, given unlimited time to grow, how many bunnies do you expect there to be in the population?

Note: No integration is required for this problem.

350

$\infty$ The population will never stop growing.

700

Explanation

The equation given is a model of logistic growth. Note that one of the other common forms this equation might be given in is $\frac{\mathrm{d} P}{\mathrm{d} t} = 5P\left ( 1-\frac{P}{350} \right )$ . What we want to know is what the population will be given unlimited time, or what $\lim_{t\rightarrow \infty} P(t)$ is.

Looking at the form of the derivative above, note that if we start with 14 bunnies, we get a positive derivative. This means that the population is increasing. As our function is continuous, the population will keep growing until the derivative hits 0. When does this occur?

In the above form, it should be more clear that the derivative is only 0 at and . Thus, the population will keep growing, but never go above 350, because if it were to hit 350, the derivative would be 0 and growth would stop. (Alternatively, if the population were above 350, you can see the derivative would be negative and that the population would shrink back down to 350. Indeed, it wasn't necessary to tell you we started with 14 bunnies -- the limit will be the same for any positive starting value.)

Asymptoteplot

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

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

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

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

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

Explanation

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

The function $f(x)=\frac{3x}{x^2-5}$ has a horizontal asymptote at y=0. What does the presence of the horizontal asymptote imply?

None of the other answers.

The function will produce only positive outputs.

At x=0 the function is undefined.

The limit of the function as x-values approach 0 tends to either positive or negative infinity.

The function is discontinuous at y=0.

Explanation

Unlike vertical asymptotes, horizontal asymptotes of certain functions may be crossed. In these cases, and in the case of the given function, the horizontal asymptote may be crossed and even have defined values laying on it (e.g., f(0)=0 for the function provided).

What is meaningful about the horizontal asymptote in this example is that it suggests the behavior of the function at large magnitudes. The denominator will increase much faster than the numerator. That is to say that the will grow exponentially larger the in the numerator such that, at large values (both positive and negative), the function will output y-values tending toward y=0.

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

The limit does not exist.

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 terms, which is $\frac{-2}{1} = -2$ .

Asymptoteplot

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

$\frac{(6x + 1)}{(8x - 17)}$

$\frac{(8x - 17)}{(6x + 1)}$

$-\frac{(6x + 1)}{(8x - 17)}$

$-\frac{(8x - 17)}{(6x + 1)}$

Explanation

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\&\text{to level out, flattening towards a certain value. This value}\&\text{is what is known as a horizontal asymptote. An asymptote is}\&\text{a value that a function approaches, but never actually reaches.}\&\text{Think of a horizontal asymptote as the limit of a function as}\&\text{x approaches infinity. In such a case, as x approaches infinity,}\&\text{any constants added or subtracted in the numerator and denominator}\&\text{become irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=0.75\&\text{This matches the ratio of coefficients for the function: }\&\frac{(6x + 1)}{(8x - 17)}\end{align*}$

Asymptoteplot

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

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

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

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

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

Explanation

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

Asymptoteplot

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

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

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

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

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

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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