Chapter 2.6, Problem 12E

Solution

The limit, $\underset{x\to 3^{+}}{\lim }f\left(x\right)$ on the basis of the graph.

The limit is $-\infty$ .

Given:

The graph of function $f\left(x\right)$ is:

  

Concept Used:

The line $x=a$ is vertical asymptote of the graph of $f\left(x\right)$ , if $\underset{x\to a^{-}}{\lim }f\left(x\right)=\pm \infty \text{ and }\underset{x\to a^{+}}{\lim }f\left(x\right)=\pm \infty$ .

Or, it can be directly intemperate the limit from graph.

Calculation:

Consider, $\underset{x\to 3^{+}}{\lim }f\left(x\right)$ .

Refer to the graph given, when $x$ approaches to $3$ from right, the value of $f\left(x\right)$ decreases to infinity.

Hence, it is clear $\underset{x\to 3^{+}}{\lim }f\left(x\right)=-\infty$

And also, from the graph the vertical asymptote is $x=3$ , so $\underset{x\to 3^{-}}{\lim }f\left(x\right)=+\infty$

Conclusion:

The limit is $-\infty$ .