the slant asymptote of $f(x)=\frac{x^{3}-2x^{2}+x - 4}{x^{2}+1}$ is $y = x - 2$. please select the best…
the slant asymptote of $f(x)=\frac{x^{3}-2x^{2}+x - 4}{x^{2}+1}$ is $y = x - 2$. please select the best answer from the choices provided. o t o f
Answer
Explanation:
Step1: Perform polynomial long - division
Divide $x^{3}-2x^{2}+x - 4$ by $x^{2}+1$. We know that when we divide a polynomial $f(x)=a_nx^n+\cdots+a_0$ by $g(x)=b_mx^m+\cdots + b_0$ ($n\geq m$) using long - division. Dividing $x^{3}-2x^{2}+x - 4$ by $x^{2}+1$: [ \begin{align*} \frac{x^{3}-2x^{2}+x - 4}{x^{2}+1}&=x - 2+\frac{-x - 2}{x^{2}+1} \end{align*} ] As $x\to\pm\infty$, the term $\frac{-x - 2}{x^{2}+1}\to0$ since the degree of the numerator ($1$) is less than the degree of the denominator ($2$).
Answer:
T