Q 7. - Calculus

Question

If $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$, $\underset{x\to \infty }{\lim }f\left(x\right)=3$, and $\underset{x\to -1^{+}}{\lim }f\left(x\right)=\infty$, what can you say about any horizontal and vertical asymptotes of $f$?

Step-by-Step Solution

Answer

The horizontal asymptote at $y=3$ and the vertical asymptote at $x=1$.

Step 1. Given information.

It is given that $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$, $\underset{x\to \infty }{\lim }f\left(x\right)=3$ and $\underset{x\to 1^{+}}{\lim }f\left(x\right)=\infty$.

Step 2. Find any horizontal and vertical asymptotes of f .

From the given expression $\underset{x\to \infty }{\lim }f\left(x\right)=3$, for any value of $x$, the functional value is $3$ i.e., $f\left(x\right)=3$ is a horizontal line.

So, $f$ has a horizontal asymptote at $y=3$.

Again, for the expression $\underset{x\to 1^{+}}{\lim }f\left(x\right)=\infty$ at $x=1$ the function have a vertical line.

So, $f$ has a vertical asymptote at $x=1$.

Step 3. Simplified answer.

The horizontal asymptote at $y=3$ and the vertical asymptote at $x=1$.