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2-71.

If $f(x)=\frac{x-3}{x+5}$, evaluate:

1. $\lim\limits _ { x \rightarrow \infty } f ( x )$

What does the graph look like to the right?

1. $\lim\limits _ { x \rightarrow - \infty } f ( x )$

What does the graph look like to the left?

1. $\lim\limits _ { x \rightarrow - 5 } f ( x )$

Since there is a vertical asymptote at $x = − 5$, the limit does not exist. But we can still determine if $f(x)$ approaches $+\infty$, $−\infty$ or both. We are to determine if there is agreement among the limits as $x \rightarrow − 5$ from each side.

$\text{Test }\lim \limits_{x\rightarrow -5^{+}}f(x)=\lim \limits_{x\rightarrow -5^{+}}\frac{x-3}{x+5}=$

Test a point close to $x = −5$ but a little bit larger:

$\lim \limits_{x\rightarrow -4.9^{+}}\frac{x-3}{x+5}=\frac{(-)}{(+)}=(-)$

Therefore, $x \rightarrow −5^+, f(x) \rightarrow −\infty$

$\text{Now test }\lim \limits_{x\rightarrow -5^{-}}f(x)$

$\text{You will find }f(x)\rightarrow +\infty .$

$\text{Since }\lim \limits_{x\rightarrow -5^{+}}f(x)\neq \lim \limits_{x\rightarrow -5^{-}}f(x)$

$\lim \limits_{x\rightarrow -5}f(x)\text{ does not exist.}$

1. $f(x − 5)$

Evaluate and simplify.

1. $f(2m + 3)$

Refer to hint (d).

1. $f(x + h)$

Refer to hint (d).

1. For parts (a) and (b), explain the graphical significance of $\lim\limits _ { x \rightarrow \infty } f ( x )$ and $\lim\limits _ { x \rightarrow - \infty } f ( x )$.

Refer to hints (a) and (b).