AP Calculus BC 1.15 Connecting Limits At Infinity and Horizontal Asymptotes Questions

Review limits at infinity by connecting end behaviour, dominant terms and horizontal asymptotes in rational functions, including how leading terms control the limit.

Covered syllabus points

LIM-2.D—Interpret the behavior of functions using limits involving infinity—Topic 1.15

[Maximum number: 2]

An invasive species of plant appears in a fruit grove at time t=0 and begins to spread. The function C defined by $C(t)=7.6 \arctan (0.2 t)$ models the number of acres in the fruit grove affected by the species t weeks after the species appears. It can be shown that $C^{\prime}(t)=\frac{38}{25+t^{2}}$ .

(Note: Your calculator should be in radian mode.)

Assume that the invasive species continues to spread according to the given model for all times t>0. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.

C Assume that the invasive species continues to spread according to the given model for all times t>0.

Write a limit expression that describes the end behavior of the rate of change in the number of acres

affected by the species. Evaluate this limit expression.

$\lim _{t \rightarrow \infty} C^{\prime}(t)=\lim _{t \rightarrow \infty} \frac{38}{25+t^{2}}$	Limit expression	Point 5 (P5)
= 0	Value	Point 6 (P6)

Scoring Notes for Part C

- P5 can be earned for either $\lim _{t \rightarrow \infty} C^{\prime}(t)$ or $\lim _{t \rightarrow \infty} C(t)$ .

- A response that includes $\lim _{t \rightarrow \infty} C(t)$ is not eligible to earn P6.

- For P6, arithmetic with infinity, e.g., $\frac{38}{25+\infty^{2}}=0$ , will be considered as scratch work and will not

be considered in scoring.

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Covered syllabus points

Question 1