CPM Homework Help : PCT Problem 8-9

Recall your work from the race to infinity. Now consider the following limits. First determine which term will dominate the function and then determine the limit. If you get stuck, you can try substituting a large value for $x$ .

$\lim\limits _ { x \rightarrow \infty } 50 \operatorname { log } x - x ^ { 2 }$
Hint (a):
As $x→∞$ , does a log graph or $x^2$ graph go up faster?

$\lim\limits _ { x \rightarrow \infty } 1.5 ^ { x } - 300 x ^ { 4 }$
Hint (b):
As $x→∞$ , does an exponential graph or $x^{4}$ graph go up faster?

$\lim\limits _ { x \rightarrow \infty } 1000 ( 2 ) ^ { x } - 3 ^ { x }$
Hint (c):
Which is steeper, $2^{x}$ or $3^{x}$ ?
Remember the coefficient ( $1000$ ) is VERY small when $x→∞$ and does not need to be considered.

$\lim\limits _ { x \rightarrow \infty } x - 30 \sqrt { x }$
Hint (d):
As $x→∞$ , does a linear graph or square root graph go up faster?