Home > APCALC > Chapter 3 > Lesson 3.2.1 > Problem3-44

3-44.

Given the function $f ( x ) = \large\frac { x ^ { 2 } + 2 x - 8 } { x ^ { 2 } - 2 x }$, calculate the following limits without using your graphing calculator.

1. $\lim\limits _ { x \rightarrow 6 } f ( x )$

Evaluate.

1. $\lim\limits _ { x \rightarrow 2 } f ( x )$

Before finding the limit, factor the numerator and the denominator. If the '$0$' in the denominator cancels out, then the limit exists.

1. $\lim\limits _ { x \rightarrow \infty } f ( x )$

For all limits as $x \rightarrow \infty$, you are looking for end behavior. For example, is there a horizontal asymptote? Or does the function approach $+\infty$ or $−\infty$ in the end? To answer this question, compare the highest power in the numerator with the highest power in the denominator.

Highest power in the numerator: $x^2$

Highest power in the denominator: $x^2$

$\lim\limits_{x\rightarrow \infty }\frac{x^{2}}{x^{^{2}}}=\lim\limits_{x\rightarrow \infty }1=1$

Note: this means that $f(x)$ has a horizontal asymptote of $y = 1$.

1. $\lim\limits _ { x \rightarrow 0 } f ( x )$

First, factor the expression. Visualize the left-hand and right-hand limits of the simplified function. Do they match up?

You can prove this by showing that$\lim\limits_{x\rightarrow 0^{-1}}f(x)\neq \lim\limits_{x\rightarrow 0^{+}}\;f(x).$

Limit does not exist.