CPM Homework Help : CALC3RD Problem 8-20

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8-20.

Evaluate each of the following limits.

$\lim\limits _ { x \rightarrow 2 } ( - \frac { 1 } { x - 2 } + \frac { x - 1 } { 2 - x } )$
Step (a):
$= \lim\limits_{x\rightarrow 2}\left ( -\frac{1}{x-2} \right )+\lim\limits_{x\rightarrow 2}\left ( \frac{x-1}{2-x} \right )$

$\lim\limits _ { x \rightarrow - 3 } ( \frac { | x + 3 | } { 2 x ^ { 2 } + 6 x + 1 } )$
Hint (b):
Factor the denominator and rewrite the numerator as a piecewise. Notice that the pieces might or might not meet at $x = 3$ .

$\lim\limits _ { x \rightarrow - \infty } ( 2 x - 3 + \frac { 5 } { x + 6 } )$
Hint (c):
What does the graph look like as $x→ -∞$ ?
What is the end behavior of $y=2x-3+\frac{5}{x+6}?$

$\lim\limits _ { x \rightarrow 4 } ( \frac { 4 - x } { 2 - \sqrt { x } } )$
Hint (d):
Before taking the limit, multiply the top and bottom of the fraction by the conjugate of the bottom.

$\lim\limits _ { x \rightarrow \infty } ( \frac { x ^ { 3 } - 5 x + 8 } { 24 x + 54 } )$
Hint (e):
$\text{Limit }→ ∞$ . Compare the highest powers of the top and the bottom. What does this graph look like at the end?

$\lim\limits _ { x \rightarrow 2 } ( \frac { 2 x ^ { 2 } - 5 x + 2 } { 5 x ^ { 2 } - 7 x - 6 } )$
Hint (f):
Before taking the limit, factor.

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