Respuesta :
Answer:
The slant asymptote is [tex]y=5 x + 5[/tex].
Step-by-step explanation:
Line [tex]y=mx+b[/tex] is a slant asymptote of the function [tex]y=f{\left(x \right)}[/tex], if either [tex]m=\lim_{x \to \infty}\left(\frac{f{\left(x \right)}}{x}\right)=L[/tex] or [tex]m=\lim_{x \to -\infty}\left(\frac{f{\left(x \right)}}{x}\right)=L[/tex], and L is finite.
We want to find the slant asymptotes of the function
[tex]f(x)=\frac{5 x^{4} + x^{2} + x}{x^{3} - x^{2} + 5}[/tex]
First, do polynomial long division
[tex]\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}[/tex]
Next, we use the above definition,
The first limit is
[tex]\lim_{x \to \infty}\left(\frac{6x^2-24x-25}{x^3-x^2+5}\right)=0[/tex]
The second limit is
[tex]\lim_{x \to -\infty}\left(\frac{6x^2-24x-25}{x^3-x^2+5}\right)=0[/tex]
The rational term approaches 0 as the variable approaches infinity.
Thus, the slant asymptote is [tex]y=5 x + 5[/tex].
