  ### Home > APCALC > Chapter 3 > Lesson 3.4.2 > Problem3-168

3-168.

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. $\lim\limits _ { x \rightarrow 0 } \large\frac { \sqrt { 3 - x } - \sqrt { 3 } } { x }$

This is Ana's definition of the derivative: $f'(a)=\lim\limits_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$

That means $f(x)=\sqrt{3-x}$ and $a = 0$.

Find $f ^\prime (x)$ and then the limit will equal $f ^\prime (0)$.

Or you could multiply the numerator and denominator by the conjugate of the numerator.

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

Evaluate. The denominator will not equal zero so the limit and the actual value agree at $x = 2$.

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

Rewrite the absolute value as a piecewise function but, since $x \rightarrow 2^+$, only consider the piece in which $x > 2$.

1. $\lim\limits _ { x \rightarrow \infty } ( e ^ { - x } + 1 )$

Think about what the graph of $y = e^{−x}$ looks like. What does it look like all the way to the right, as $x \rightarrow \infty$? How does the $+1$ affect the graph?