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

3-123.

Evaluate each limit. If the limit does not exist, say so but also state if $y$ is approaching positive or negative infinity.

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

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

1. $\lim\limits_ { x \rightarrow 4 } \frac { 2 - \sqrt { x } } { x - 4 }$

$-\lim\limits_{x\rightarrow 4}\left (\frac{\sqrt{x}-2}{x-4} \right )$
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. $\lim\limits_ { x \rightarrow 6 } \frac { 2 x ^ { 2 } - 12 x } { x ^ { 2 } + x - 42 }$

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

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

Refer to the hint in part (a).