Question

Use the graph of $f$to find the following limits.

$\left(i\right)\underset{x\to \infty }{\lim }f\left(x\right)$

$\left(ii\right)\underset{x\to -\infty }{\lim }f\left(x\right)$

Short answer

The value of limit of the given function is

$\begin{array}{c}\text{(i)}\underset{x\to \infty }{\lim }f\left(x\right)=4\\ \left(ii\right)\underset{x\to -\infty }{\lim }f\left(x\right)=-2\end{array}$

Step-by-step solution

Step 1. Given information.

Here, the limit of the function is given:

$\begin{array}{l}\left(i\right)\underset{x\to \infty }{\lim }f\left(x\right)\\ \left(ii\right)\underset{x\to -\infty }{\lim }f\left(x\right)\end{array}$

Step 2. Concept  used.

Here we use the concept of limit at infinity.

Step 3.(i) Simplification.

The given graph is:

As we look to the far right,

We see that for numerically large value of $x$, the value of $f\left(x\right)$are close to $4$.

(ii) Simplification.

By the given graph. We see that:

As we look, we see that for numerically large value of $x$, the value of $f\left(x\right)$are close to $-2$.

Therefore, our final answer is $\text{4,-2.}$

(b) State the equation of the horizontal asymptote.

The equation of the given graph is:

$\begin{array}{c}y=4\\ y=-2\end{array}$

Step 1. Given information.

Here, the graph is given:

Step 2. Concept  used.

Here we use the concept of horizontal asymptote.

Step 3.(i) Simplification.

The given graph is:

As we look,

We see that the curve illustrated the graph has the line $y=4,y=-2$ as a horizontal asymptote.

Therefore, our final answer is $\text{4,-2.}$