# How do you find the end behavior of y = (2x+3) /( x+2)?

May 10, 2015

The dominant terms in the numerator and denominator as $x \to \pm \infty$, will be the ones in $2 x$ and $x$ respectively.

The limit of $\frac{2 x}{x}$ as $x \to \infty \mathmr{and} - \infty$ is 2.

More formally, we have

$y = \frac{2 x + 3}{x + 2} = \frac{2 x + 4 - 1}{x + 2} = \frac{2 \left(x + 2\right) - 1}{x + 2}$

$= 2 \frac{x + 2}{x + 2} - \frac{1}{x + 2} = 2 - \frac{1}{x + 2}$

The term $\frac{1}{x + 2} \to 0$ as $x \to \pm \infty$.