CPM Homework Help : CALC Problem 3-123

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3-123.

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

$\lim\limits_ { x \rightarrow \infty } \frac { x ^ { 3 } - x ^ { - 3 } } { 5 x ^ { 3 } + x ^ { - 3 } }$
Hint (a):
As with all limits where $x→∞$ or $x→−∞$ , compare the highest-power term in the numerator with the highest-power term in the denominator. Be sure to consider coefficients.

$\lim\limits_ { x \rightarrow 4 } \frac { 2 - \sqrt { x } } { x - 4 }$
Hint 1(b):
$-\lim\limits_{x\rightarrow 4}\left (\frac{\sqrt{x}-2}{x-4} \right )$
This is Ana's method to find a derivative. If you need more guidance, refer to the hints in problem 3-110 part (b).
Hint 2(b):
Or you could multiply the numerator and the denominator by the conjugate of $(2-\sqrt{x}).$

$\lim\limits_ { x \rightarrow 6 } \frac { 2 x ^ { 2 } - 12 x } { x ^ { 2 } + x - 42 }$
Hint (c):
When you evaluate at $x = 6$ , there is a $0$ in the denominator. Perhaps you can cancel it out. Try factoring.

$\lim\limits_ { x \rightarrow - \infty } \frac { x ^ { 3 } } { 4 + x ^ { 2 } }$
Hint (d):
Refer to the hint in part (a).

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