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Step-by-step Answer

PAGE: 60

SET: Exercises

PROBLEM: 9

Please look in your text book for this problem Statement

(a)

Find $\small \lim_{x\rightarrow-7}f(x)$ .

Observe the graph.

Left hand limit $\small \lim_{x\rightarrow-7^-}f(x)$ :

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

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

Right hand limit $\small \lim_{x\rightarrow-7^+}f(x)$ :

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, $\small \lim_{x\rightarrow-7}f(x)$ is exist.

$\small \lim_{x\rightarrow-7}f(x)=-\infty$ .

(b)

Find $\small \lim_{x\rightarrow-3}f(x)$ .

Observe the graph.

Left hand limit $\small \lim_{x\rightarrow-3^-}f(x)$ :

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

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

Right hand limit $\small \lim_{x\rightarrow-3^+}f(x)$ :

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, $\small \lim_{x\rightarrow-3}f(x)$ is exist.

$\small \lim_{x\rightarrow-3}f(x)=\infty$ .

(c)

Find $\small \lim_{x\rightarrow0}f(x)$ .

Observe the graph.

Left hand limit $\small \lim_{x\rightarrow0^-}f(x)$ :

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

So $\small \lim_{x\rightarrow0^-}f(x)=\infty$ .

Right hand limit $\small \lim_{x\rightarrow0^+}f(x)$ :

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, $\small \lim_{x\rightarrow0}f(x)$ is exist.

$\small \lim_{x\rightarrow0}f(x)=\infty$ .

(d)

Find $\small \lim_{x\rightarrow6^-}f(x)$ .

Observe the graph.

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

So $\small \lim_{x\rightarrow6^-}f(x)=-\infty$ .

(e)

Find $\small \lim_{x\rightarrow6^+}f(x)$ .

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

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

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

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 $\small \lim_{x\rightarrow6^-}f(x)=-\infty$ .

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

The vertical asymptotes are $\small x=-7$ , $\small x=-3$ , $\small x=0$ and $\small x=6$ .

(a) $\small \lim_{x\rightarrow-7}f(x)=-\infty$ .

(b) $\small \lim_{x\rightarrow-3}f(x)=\infty$ .

(d) $\small \lim_{x\rightarrow6^-}f(x)=-\infty$ .

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

(f) The vertical asymptotes are $\small x=-7$ , $\small x=-3$ , $\small x=0$ and $\small x=6$ .

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