CPM Homework Help : APCALC Problem 10-125

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10-125.

Evaluate each of the following limits.

$\lim\limits _ { x \rightarrow 1 ^ { + } } \frac { \operatorname { ln } ( x ) } { x - 1 }$
Hint (a):
Use one iteration of l'Hôpital's Rule.

$\lim\limits _ { x \rightarrow 0 } \frac { e ^ { x } - 1 } { \operatorname { cos } ( x ) - 1 }$
Hint (b):
Use one iteration of l'Hôpital's Rule.

$\lim\limits _ { x \rightarrow 0 ^ { + } } x \operatorname { ln } ( x )$
Step (c):
Rewrite this as:
$\lim\limits_{x\to 0^+}\frac{\ln(x)}{1/x}$
Then use l'Hôpital's Rule.

$\lim\limits _ { x \rightarrow 8 } \frac { x ^ { 1 / 3 } - 2 } { x - 8 }$
Hint (d):
Factor the denominator as a difference of cubes:
$x^3 - a^3 = (x - a)(x^2 + ax - a^2)$

$\lim\limits _ { x \rightarrow 0 } \frac { x - \operatorname { tan } ^ { - 1 } ( x ) } { x ^ { 2 } }$
Hint (e):
Use two iterations of l'Hôpital's Rule.

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