Question

describe the right end behavior of the function f(x) = (40x^7 + 3x^2)/(16x^6 - 2x) by finding lim(x→∞) f(x).
lim(x→∞) f(x) = (simplify your answer.)

Explanation:

Step1: Divide numerator and denominator by highest - power of x

Divide both the numerator and denominator of $f(x)=\frac{40x^{7}+3x^{2}}{16x^{6}-2x}$ by $x^{6}$. We get $\lim_{x
ightarrow\infty}\frac{40x + \frac{3}{x^{4}}}{16-\frac{2}{x^{5}}}$.

Step2: Apply limit rules

As $x
ightarrow\infty$, $\lim_{x
ightarrow\infty}\frac{3}{x^{4}} = 0$ and $\lim_{x
ightarrow\infty}\frac{2}{x^{5}}=0$. So, $\lim_{x
ightarrow\infty}\frac{40x+\frac{3}{x^{4}}}{16 - \frac{2}{x^{5}}}=\lim_{x
ightarrow\infty}\frac{40x+0}{16 - 0}$.

Step3: Evaluate the limit

$\lim_{x
ightarrow\infty}\frac{40x}{16}=\infty$.

Answer:

$\infty$