Calcolatore di asintoti

Your input: find the vertical, horizontal and slant asymptotes of the function $$$f(x)=\frac{x^{2} - 7}{x - 4}$$$

The line $$$x=L$$$ is a vertical asymptote of the function $$$y=\frac{x^{2} - 7}{x - 4}$$$, if the limit of the function (one-sided) at this point is infinite.

In other words, it means that possible points are points where the denominator equals $$$0$$$ or doesn't exist.

So, find the points where the denominator equals $$$0$$$ and check them.

$$$x=4$$$, check:

$$$\lim_{x \to 4^+}\left(\frac{x^{2} - 7}{x - 4}\right)=\infty$$$ (for steps, see limit calculator).

Since the limit is infinite, then $$$x=4$$$ is a vertical asymptote.

Line $$$y=L$$$ is a horizontal asymptote of the function $$$y=f{\left(x \right)}$$$, if either $$$\lim_{x \to \infty} f{\left(x \right)}=L$$$ or $$$\lim_{x \to -\infty} f{\left(x \right)}=L$$$, and $$$L$$$ is finite.

Calculate the limits:

$$$\lim_{x \to \infty}\left(\frac{x^{2} - 7}{x - 4}\right)=\infty$$$ (for steps, see limit calculator).

$$$\lim_{x \to -\infty}\left(\frac{x^{2} - 7}{x - 4}\right)=-\infty$$$ (for steps, see limit calculator).

Thus, there are no horizontal asymptotes.

Do polynomial long division $$$\frac{x^{2} - 7}{x - 4}=x + 4 + \frac{9}{x - 4}$$$ (for steps, see polynomial long division calculator).

The rational term approaches 0 as the variable approaches infinity.

Thus, the slant asymptote is $$$y=x + 4$$$.

Answer

Vertical asymptote: $$$x=4$$$

No horizontal asymptotes.

Slant asymptote: $$$y=x + 4$$$