AP Calculus BC : 1.15 Connecting Limits at Infinity and Horizontal Asymptotes- Exam Style questions with Answer- MCQ

Question

Let f be the function defined by \(f(x)=\frac{3x+5}{x+2}\). Which of the following statements are true?

I .The graph of f has a horizontal asymptote at y=3 because .\(\lim_{x\rightarrow \infty }f(x)=3\).
II. The graph of f has a horizontal asymptote at y=3 because \(\lim_{x\rightarrow -\infty }f(x)=3\)
III. The graph of f has a vertical asymptote at x=−2 because \(\lim_{x\rightarrow2^{+}}f(x)=-\infty \).
A I only
B III only
C I and II only
D I, II, and III

Answer/Explanation

Ans:D

Question

The population on an island is modeled by for t≥0, where P(t) is the number of people on the island after t years. What is  ?
A 100
B \(\frac{500}{3}\)
C 250
D 5000

Answer/Explanation

Ans:C

Question

Let f be the function defined by \(f(x)=\frac{5x^{20}}{8e^{x}+9x^{20}}\) for x>0. Which of the following is a horizontal asymptote to the graph of f ?
A y=0
B y=\(\frac{5}{9}\)
C y=\(\frac{5}{8}\)
D There is no horizontal asymptote to the graph of f.

Answer/Explanation

Ans:A

Question 

 Find the horizontal asymptote(s) of \(f(t)=\frac{27t-18}{3t+8}\)
(A) y = 9
(B) y = 6
(C)\(  y=-\frac{9}{4}\)
(D) There are no horizontal asymptotes.

Answer/Explanation

Ans:(A)

A) To find the horizontal asymptote(s), you find the limits of the function as t → ∞ and t → -∞.

\(\lim_{t\rightarrow \infty }\frac{27t-18}{3t+8}=\lim_{t\rightarrow \infty }\frac{27-\frac{18}{t}}{3+\frac{8}{t}}=\frac{27}{3}=9\)
Likewise,

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