# How do you find the asymptotes for f(x) = sqrt(x^2+5) - x?

Feb 6, 2017

Horizontal asymptote : $y = 0 \leftarrow$ See illustrative graph.

#### Explanation:

$f = \sqrt{{x}^{2} + 5} - x = \frac{\left(\sqrt{{x}^{2} + 5} - x\right) \left(\sqrt{{x}^{2} + 5} + x\right)}{\sqrt{{x}^{2} + 5} + x}$

$= \frac{{x}^{2} + 5 - {x}^{2}}{\sqrt{{x}^{2} + 5} - x} + \frac{5}{\sqrt{{x}^{2} + 5} + x}$

$\to 0$, as $x \to \infty$

So, y = 0 is the asymptote, .

graph{(sqrt(x^2+5)-x-y)y=0 [-10, 10, -5, 5]}