CPM Homework Help : APCALC Problem 9-112

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9-112.

No calculator! Evaluate the following limits.

$\lim\limits _ { x \rightarrow \infty } \frac { 3 x ^ { 5 } - 2 x ^ { 2 } + 1 } { 2 x ^ { 6 } + 4 x - 1 }$
Hint (a):
What is the dominant term in the numerator? What is the dominant term in the denominator? How do these terms compare?

$\lim\limits _ { x \rightarrow - \infty } e ^ { - 2 x }$
Hint (b):
Graph $y = e^{–2x}$ . What happens to the graph as $x → -∞$ ?

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { cot } ( x ) } { \operatorname { ln } ( x ) }$
Hint (c):
l'Hôpital's Rule can be used here. Or, you can graph the function.

$\lim\limits _ { n \rightarrow \infty } ( \displaystyle\sum _ { k = 1 } ^ { n } ( \frac { 1 } { 3 } ) ^ { k } )$
Hint (d):
This is the geometric series:
$\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...$

Compute without a calculator

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