CPM Homework Help : CALC Problem 3-168

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3-168.

Evaluate each limit. If the limit does not exist, say so but also state if $y$ is approaching positive or negative infinity.

$\lim\limits_ { x \rightarrow 0 } \frac { \sqrt { 3 - x } - \sqrt { 3 } } { x }$
Hint 1(a):
This is Ana's definition of the derivative:
$f'(a)=\lim\limits_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$
That means $f(x)=\sqrt{3-x}\text{ and }a=0.$
Find $f^\prime(x)$ and then the limit will equal $f^\prime(0)$ .
Hint 2(a):
Or you could multiply the numerator and denominator by the conjugate of the numerator.

$\lim\limits_ { x \rightarrow 2 } \frac { x + 3 } { x - 4 }$
Hint (b):
Evaluate. The denominator will not equal zero so the limit and the actual value agree at $x = 2$ .

$\lim\limits_ { x \rightarrow 2 ^ { + } } \frac { x ^ { 2 } | x - 2 | } { x - 2 }$
Hint (c):
Rewrite the absolute value as a piecewise function but, since $x → 2^+$ , only consider the piece in which $x > 2$ .

$\lim\limits_ { x \rightarrow \infty } e ^ { - x } + 1$
Hint (d):
Think about what the graph of $y = e^{−x}$ looks like. What does it look like all the way to the right, as $x → ∞$ ? How does the $+1$ affect the graph?

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