  ### Home > PC > Chapter 8 > Lesson 8.1.2 > Problem8-24

8-24.

Evaluate the following limits. (Note the negative sign in parts (a), (b), and (c).)

1. $\lim\limits _ { x \rightarrow - \infty } \frac { 2 x } { x ^ { 3 } }$

The dominant term is in the denominator.
As $x→∞$, the fraction gets closer to $0$.

1. $\lim\limits _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 5 } } { 2 x }$

The dominant term is $\left|x\right|$ in the numerator (square root of $x^{2}$ is $\left|x\right|$) and $x$ in the denominator, so the answer is the ratio of the coefficients.

$\text{i.e. }\frac{1}{-2}$

1. $\lim\limits _ { x \rightarrow - \infty } ( \operatorname { cos } x )$

The cosine function oscillates between $1$ and $−1$ forever. Since a limit must approach a single height, the limit does not exist.

1. $\lim\limits _ { x \rightarrow 0 } \frac { | x | } { x }$

Is the graph approaching the SAME height from both sides?

1. $\lim\limits _ { x \rightarrow 0 ^ { + } } \operatorname { log } _ { 2 } x$

What height is the graph approaching as $x→0^{+}$?