# How do you find intercepts, extrema, points of inflections, asymptotes and graph f(x)=x/sqrt(x^2+7)?

May 7, 2018

See below.

#### Explanation:

Both intercepts are $0$.

$\sqrt{{x}^{2} + 7} > 0$ for all $x$, so there are no vertical asymptotes.

${\lim}_{x \rightarrow \infty} f \left(x\right) = {\lim}_{x \rightarrow \infty} \frac{x}{x \sqrt{1 + \frac{7}{x} ^ 2}} = 1$, so $y = 1$ is a horizontal asymptote on the right.

${\lim}_{x \rightarrow - \infty} f \left(x\right) = {\lim}_{x \rightarrow - \infty} \frac{x}{- x \sqrt{1 + \frac{7}{x} ^ 2}} = - 1$, so $y = - 1$ is a horizontal asymptote on the left.

$f ' \left(x\right) = \frac{7}{{x}^{2} + 7} ^ \left(\frac{3}{2}\right) > 0$ for all $x$, so there are no local extrema.

$f ' ' \left(x\right) = \frac{- 21 x}{{x}^{2} + 7} ^ \left(\frac{5}{2}\right)$ changes sign at $\left(0 , 0\right)$, so I have been taught that $\left(0 , 0\right)$ is an inflection point. (Note that: $f ' \left(0\right) \ne 0$, so some would say $\left(0 , 0\right)$ is not an inflection point.)

graph{x/sqrt(x^2+7) [-16.01, 16.03, -7.95, 8.05]}