  ### Home > APCALC > Chapter 3 > Lesson 3.3.3 > Problem3-123

3-123.

Evaluate each limit. If the limit does not exist due to a vertical asymptote, then add an approach statement stating if $y$ is approaching negative or positive infinity.

1. $\lim\limits _ { x \rightarrow \infty } \large\frac { x ^ { 3 } - x ^ { - 3 } } { 5 x ^ { 3 } + x ^ { - 3 } }$

As with all limits where $x \rightarrow \infty$ or $x \rightarrow −\infty$,compare the highest-power term in the numerator with the highest-power term in the denominator. Be sure to consider coefficients.

1. $\lim\limits _ { x \rightarrow 4 } \large\frac { 2 - \sqrt { x } } { x - 4 }$

$-\lim\limits_{x\rightarrow 4}\left (\frac{\sqrt{x}-2}{x-4} \right )$

This is Ana's method to find a derivative. If you need more guidance, refer to the hints in problem 3-110 part (b).

Or you could multiply the numerator and the denominator by the conjugate of $(2-\sqrt{x})$.

1. $\lim\limits _ { x \rightarrow 6 } \large\frac { 2 x ^ { 2 } - 12 x } { x ^ { 2 } + x - 42 }$

When you evaluate at $x = 6$, there is a $0$ in the denominator. Perhaps you can cancel it out. Try factoring.

1. $\lim\limits _ { x \rightarrow - \infty } \large\frac { x ^ { 3 } } { 4 + x ^ { 2 } }$

Refer to the hint in part (a).