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(a)

Find $\"\\small$ .

Observe the graph.

Left hand limit $\"\\lim_{x\\rightarrow2^-}R(x)\"$ :

As $\"x\"$ tends to $\"2\"$ from left side, $\"R(x)\"$ approaches to negative large number.

$\"\\small$ .

Right hand limit $\"\\lim_{x\\rightarrow2^+}R(x)\"$ :

As $\"x\"$ tends to $\"2\"$ from right side, $\"R(x)\"$ approaches to negative large number.

So $\"\\lim_{x\\rightarrow2^+}R(x)=-\\infty\"$ .

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

$\"\\small$ .

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(b)

Find $\"\\small$ .

Observe the graph.

Left hand limit $\"\\lim_{x\\rightarrow5^-}R(x)\"$ :

As $\"x\"$ tends to $\"5\"$ from left side, $\"R(x)\"$ approaches to positive large number.

$\"\\small$ .

Right hand limit $\"\\lim_{x\\rightarrow5^+}R(x)\"$ :

As $\"x\"$ tends to $\"5\"$ from right side, $\"R(x)\"$ approaches to positive large number.

$\"\\lim_{x\\rightarrow5^+}R(x)=\\infty\"$ .

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

$\"\\small$ .

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(c)

Find $\"\\lim_{x\\rightarrow-3^-}R(x)\"$ .

Observe the graph.

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

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

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(d)

Find $\"\\small$ .

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

So $\"\\small$ .

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(e)

Find the vertical asymptote.

Observe the graph.

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

So $\"\\small$ .

As $\"x\"$ approaches to $\"2\"$ from left side, $\"R(x)\"$ approaches to negative large number.

So $\"\\small$ .

As $\"x\"$ approaches to $\"5\"$ from right side, $\"R(x)\"$ approaches to positive large number.

So $\"\\lim_{x\\rightarrow5^+}R(x)=\\infty\"$ .

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

$\"\"$

(a) $\"\\small$ .

(b) $\"\\small$ .

(c) $\"\\lim_{x\\rightarrow-3^-}R(x)=-\\infty\"$ .

(d) $\"\\small$ .

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