Question

Find an equation of the slant asymptote. Do not sketch the curve. y = 5x4 + x2 + x x3 − x2 + 5

Answer 1

Answer:

The slant asymptote is $y=5 x + 5$.

Step-by-step explanation:

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

We want to find the slant asymptotes of the function

$f(x)=\frac{5 x^{4} + x^{2} + x}{x^{3} - x^{2} + 5}$

First, do polynomial long division

$\frac{5 x^{4} + x^{2} + x}{x^{3} - x^{2} + 5}=5 x + 5+\frac{6 x^{2} - 24 x - 25}{x^{3} - x^{2} + 5}$

Next, we use the above definition,

The first limit is

$\lim_{x \to \infty}\left(\frac{6x^2-24x-25}{x^3-x^2+5}\right)=0$

The second limit is

$\lim_{x \to -\infty}\left(\frac{6x^2-24x-25}{x^3-x^2+5}\right)=0$

The rational term approaches 0 as the variable approaches infinity.

Thus, the slant asymptote is $y=5 x + 5$.