Home / AP Calculus AB and BC: Chapter 1 – Limits and Continuity : 1.5 – Limits and Asymptotes Study Notes

AP Calculus AB and BC: Chapter 1 – Limits and Continuity : 1.5 – Limits and Asymptotes Study Notes

The statement \(\underset{x\rightarrow c}{lim}f\left ( x \right )=\infty \) means that f (x) approaches infinity as x approaches c.

The statement \(\underset{x\rightarrow c}{lim}f\left ( x \right )=\infty \) means that f (x) approaches negative infinity as x approaches c.

Definition of Vertical Asymptote

If f (x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f.

The graph of rational function given by \(y=\frac{f\left ( x \right )}{g\left ( x \right )}\) has a vertical asymptote at x = c if \(g\left ( c \right )=0\) and \(f\left ( c \right )\neq 0\).

Definition of Horizontal Asymptote

A line y = b is a horizontal asymptote of the graph of a function y = f (x) if either

\(\underset{x\rightarrow \infty}{lim}f\left ( x \right )=b\) or \(\underset{x\rightarrow -\infty}{lim}f\left ( x \right )=b\).

The statement \(\underset{x\rightarrow \infty}{lim}f\left ( x \right )=L\) means that f (x) has the limit L as x approaches infinity.

The statement \(\underset{x\rightarrow -\infty}{lim}f\left ( x \right )=L\) means that f (x) has the limit L as x approaches negative infinity.

Example 1 

  • Find all vertical asymptotes of the graph of each function.

(a) \(f\left ( x \right )=\frac{x}{x^{2}-1}\)

(b) \(f\left ( x \right )=\frac{x^{2}-4x-5}{x^{2}-x-2}\)

▶️Answer/Explanation

Solution

(a) \(f\left ( x \right )=\frac{x}{x^{2}-1}\)

      \(\frac{x}{\left ( x+1 \right )\left ( x-1 \right )}\)

      Factor the denominator.

      The denominator is 0 at x = −1 and x = 1 . The numerator is not 0 at these two points. There are two vertical asymptotes, x = −1 and x = 1 .

(b)

At all values other than x = −1, the graph of f coincides with the graph of \(y=\left ( x-5 \right )/\left ( x-2 \right ).\) So, x = 2 is the only vertical asymptote. At x = −1 the graph is not continuous.

The statement \(\underset{x\rightarrow c}{lim}f\left ( x \right )=\infty \) means that f (x) approaches infinity as x approaches c.

The statement \(\underset{x\rightarrow c}{lim}f\left ( x \right )=\infty \) means that f (x) approaches negative infinity as x approaches c.

Definition of Vertical Asymptote

If f (x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f.

The graph of rational function given by \(y=\frac{f\left ( x \right )}{g\left ( x \right )}\) has a vertical asymptote at x = c if \(g\left ( c \right )=0\) and \(f\left ( c \right )\neq 0\).

Definition of Horizontal Asymptote

A line y = b is a horizontal asymptote of the graph of a function y = f (x) if either

\(\underset{x\rightarrow \infty}{lim}f\left ( x \right )=b\) or \(\underset{x\rightarrow -\infty}{lim}f\left ( x \right )=b\).

The statement \(\underset{x\rightarrow \infty}{lim}f\left ( x \right )=L\) means that f (x) has the limit L as x approaches infinity.

The statement \(\underset{x\rightarrow -\infty}{lim}f\left ( x \right )=L\) means that f (x) has the limit L as x approaches negative infinity.

Example 1 

  • Find all vertical asymptotes of the graph of each function.

(a) \(f\left ( x \right )=\frac{x}{x^{2}-1}\)

(b) \(f\left ( x \right )=\frac{x^{2}-4x-5}{x^{2}-x-2}\)

▶️Answer/Explanation

Solution

(a) \(f\left ( x \right )=\frac{x}{x^{2}-1}\)

      \(\frac{x}{\left ( x+1 \right )\left ( x-1 \right )}\)

      Factor the denominator.

      The denominator is 0 at x = −1 and x = 1 . The numerator is not 0 at these two points. There are two vertical asymptotes, x = −1 and x = 1 .

(b)

At all values other than x = −1, the graph of f coincides with the graph of \(y=\left ( x-5 \right )/\left ( x-2 \right ).\) So, x = 2 is the only vertical asymptote. At x = −1 the graph is not continuous.

Example 2 

  • Find the limit.

(a) \(\underset{x\rightarrow \infty }{lim}\frac{x^{3}-4x^{2}+7}{2x^{3}-3x-5}\)

(b) \(\underset{x\rightarrow \infty }{lim}\frac{\sqrt{4x^{2}+6x}}{3x-2}\)

▶️Answer/Explanation

Solution

Example 3

  •  Find the horizontal asymptotes of the graph of the function \(f\left ( x \right )=\frac{\sqrt[3]{2x^{3}-9}}{x}\).
▶️Answer/Explanation

Solution

Exercises – Limits and Asymptotes
Multiple Choice Questions

Question

  • \(\underset{x\rightarrow \infty }{lim}\frac{3+2x^{2}-x^{4}}{3x^{4}-5}=\)

(A) -2               (B) \(-\frac{1}{3}\)                (C) \(\frac{1}{5}\)               (D) 1

▶️Answer/Explanation

Ans:

1. B

Question

  •  What is \(\underset{x\rightarrow \infty }{lim}\frac{x^{3}+x-8}{2x^{3}+3x-1}=\)

(A) \(-\frac{1}{2}\)               (B) 0                (C) \(\frac{1}{2}\)               (D) 2

▶️Answer/Explanation

Ans:

2. C

Question

  • Which of the following lines is an asymptote of the graph of \(f\left ( x \right )=\frac{x^{2}+5x+6}{x^{2}-x-12}?\)

I. x = −3

II. x = 4

III. y = 1

(A) II only               (B) III only                (C) II and III only               (D) I, II, and III

▶️Answer/Explanation

Ans:

3. C

Question

  •  If the horizontal line y = 1 is an asymptote for the graph of the function f , which of the following statements must be true?

(A) \(\underset{x\rightarrow \infty }{lim}f\left ( x \right )=1\)

(B) \(\underset{x\rightarrow 1}{lim}f\left ( x \right )=\infty\)

(C) f (1) is undefined

(D) f (x) = 1 for all

▶️Answer/Explanation

Ans:

4. A

Question

  • If x = 1 is the vertical asymptote and y = −3 is the horizontal asymptote for the graph of the function f , which of the following could be the equation of the curve?

(A) \(f\left ( x \right )=\frac{-3x^{2}}{x-1}\)

(B) \(f\left ( x \right )=\frac{-3\left ( x-1 \right )}{x+3}\)

(C) \(f\left ( x \right )=\frac{-3\left ( x^{2}-1 \right )}{x-1}\)

(D) \(f\left ( x \right )=\frac{-3\left ( x^{2}-1 \right )}{\left (x-1 \right )^{2}}\)

▶️Answer/Explanation

Ans:

5. D

Question

  •  What are all horizontal asymptotes of the graph of \(y=\frac{6+3e^{x}}{3-3e^{x}}\) in the xy – plane?

(A) y = −1 only

(B) y = 2 only

(C) y = −1 and y = 2

(D) y = 0 and y = 2

▶️Answer/Explanation

Ans:

6. C

Question

  • Let \(f\left ( x \right )=\frac{3x-1}{x^{3}-8}.\)

(A) Find the vertical asymptote(s) of f . Show the work that leads to your answer.

(B) Find the horizontal asymptote(s) of f . Show the work that leads to your answer.

▶️Answer/Explanation

Ans:

7. (a) x = 2

    (b) y = 0

Question

  •  Let \(f\left ( x \right )=\frac{sin x}{x^{2}+2x}.\)

(A) Find the vertical asymptote(s) of f . Show the work that leads to your answer.

(B) Find the horizontal asymptote(s) of f . Show the work that leads to your answer.

▶️Answer/Explanation

Ans:

8. (a) x = −2

     (b) y = 0

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