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2-71.

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

$\lim\limits_ { x \rightarrow \infty } f ( x )$
Hint (a):
What does the graph look like to the right?

$\lim\limits _ { x \rightarrow - \infty } f ( x )$
Hint (b):
What does the graph look like to the left?

$\lim\limits_ { x \rightarrow - 5 } f ( x )$
Hint (c):
Since there is a vertical asymptote at $x = − 5$ , the limit does not exist. But we can still determine if $f(x)$ approaches $+∞$ , $−∞$ or both. We are to determine if there is agreement among the limits as $x → − 5$ from each side.
Step 1 (c):
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, $x → −5^+$ , $f(x) → −∞$
Step 2 (c):
Now test $\lim\limits_{x\rightarrow -5^{-}}f(x)$
You will find $f(x)\rightarrow +\infty$ .
Step 3 (c):
Since $\lim\limits_{x\rightarrow -5^{+}}f(x)\neq \lim\limits_{x\rightarrow -5^{-}}f(x)$ $\lim\limits_{x\rightarrow -5}f(x)$ does not exist.

$f(x − 5)$
Hint (d):
Evaluate and simplify.

$f(2m + 3)$
Hint (e):
Refer to hint (d).

$f(x + h)$
Hint (f):
Refer to hint (d).

For parts (a) and (b), explain the graphical significance of $\lim\limits_ { x \rightarrow \infty } f ( x )$ and $\lim\limits_ { x \rightarrow - \infty } f ( x )$ .
Hint (g):
Refer to hints (a) and (b).

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