Chapter 2.2, Problem 60E

To determine

Tocheck:Whether the statement “If $f\left(x\right)$ has a vertical asymptote at $x=c$ , then either $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=\infty$ or $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=-\infty$ ” is true or not.

Answer to Problem 60E

The statement “If $f\left(x\right)$ has a vertical asymptote at $x=c$ , then either $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=\infty$ or $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=-\infty$ ” is false.

Explanation of Solution

Given information: The given statement is “If $f\left(x\right)$ has a vertical asymptote at $x=c$ , then either $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=\infty$ or $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=-\infty$ ”.

Definition says that if $f\left(x\right)$ has a vertical asymptote at $x=c$ , then either $\underset{x\to c^{-}}{\lim }f\left(x\right)=\pm \infty$ or $\underset{x\to c^{+}}{\lim }f\left(x\right)=\pm \infty$ .

Both the limits need not to be equal.

Therefore, the statement “If $f\left(x\right)$ has a vertical asymptote at $x=c$ , then either $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=\infty$ or $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=-\infty$ ” is false.