(a)

Find $\"\\small$ .

Observe the graph.

Left hand limit $\"\\small$ :

As $\"x\"$ tends to $\"-7\"$ from left side, $\"f(x)\"$ approaches to negative large number.

So $\"\\small$ .

Right hand limit $\"\\small$ :

As $\"x\"$ tends to $\"-7\"$ from right side, $\"f(x)\"$ approaches to negative large number.

So $\"\\lim_{x\\rightarrow-7^+}f(x)=-\\infty\"$ .

Left hand limit and right hand limit are equal, $\"\\small$ is exist.

$\"\\small$ .

(b)

Find $\"\\small$ .

Observe the graph.

Left hand limit $\"\\small$ :

As $\"x\"$ tends to $\"-3\"$ from left side, $\"f(x)\"$ approaches to positive large number.

So $\"\\small$ .

Right hand limit $\"\\small$ :

As $\"x\"$ tends to $\"-3\"$ from right side, $\"f(x)\"$ approaches to positive large number.

So $\"\\lim_{x\\rightarrow-3^+}f(x)=\\infty\"$ .

Left hand limit and right hand limit are equal, $\"\\small$ is exist.

$\"\\small$ .

(c)

Find $\"\\small$ .

Observe the graph.

Left hand limit $\"\\small$ :

As $\"x\"$ tends to $\"0\"$ from left side, $\"f(x)\"$ approaches to positive large number.

So $\"\\small$ .

Right hand limit $\"\\small$ :

As $\"x\"$ tends to $\"0\"$ from right side, $\"f(x)\"$ approaches to positive large number.

So $\"\\lim_{x\\rightarrow0^+}f(x)=\\infty\"$ .

Left hand limit and right hand limit are equal, $\"\\small$ is exist.

$\"\\small$ .

(d)

Find $\"\\small$ .

Observe the graph.

As $\"x\"$ tends to $\"6\"$ from left side, $\"f(x)\"$ approaches to negative large number.

So $\"\\small$ .

(e)

Find $\"\\small$ .

Observe the graph.

As $\"x\"$ tends to $\"6\"$ from right side, $\"f(x)\"$ approaches to positive large number.

So $\"\\lim_{x\\rightarrow6^+}f(x)=\\infty\"$ .

(f)

Find the vertical asymptote.

Vertical asymptote :

The vertical asymptote is a line equation, toward which a function $\"f(x)\"$ approaches to either positive or negative infinity.

Observe the graph.

As $\"x\"$ tends to $\"-7\"$ from left side, $\"f(x)\"$ approaches to negative large number.

So $\"\\small$ .

As $\"x\"$ tends to $\"-3\"$ from left side, $\"f(x)\"$ approaches to positive large number.

So $\"\\small$ .

As $\"x\"$ tends to $\"0\"$ from right side, $\"f(x)\"$ approaches to positive large number.

So $\"\\lim_{x\\rightarrow0^+}f(x)=\\infty\"$ .

As $\"x\"$ tends to $\"6\"$ from left side, $\"f(x)\"$ approaches to negative large number.

So $\"\\small$ .

From the above statements, it satisfies the definition of vertical asymptotes.

The vertical asymptotes are $\"\\small$ , $\"\\small$ , $\"\\small$ and $\"\\small$ .

(a) $\"\\small$ .

(b) $\"\\small$ .

(c) $\"\\small$ .

(d) $\"\\small$ .

(e) $\"\\lim_{x\\rightarrow6^+}f(x)=\\infty\"$ .

(f) The vertical asymptotes are $\"\\small$ , $\"\\small$ , $\"\\small$ and $\"\\small$ .