Question

find the limit of the rational function (a) as x→∞ and (b) as x→ - ∞. write ∞ or - ∞ where appropriate. f(x)=\frac{4x^{6}+4x^{5}+8}{4x^{7}} lim f(x)=□ x→∞ (simplify your answer.)

Explanation:

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

Divide each term in the numerator and denominator of $f(x)=\frac{4x^{6}+4x^{5}+8}{4x^{7}}$ by $x^{7}$. We get $\frac{\frac{4x^{6}}{x^{7}}+\frac{4x^{5}}{x^{7}}+\frac{8}{x^{7}}}{\frac{4x^{7}}{x^{7}}}=\frac{\frac{4}{x}+\frac{4}{x^{2}}+\frac{8}{x^{7}}}{4}$.

Step2: Use limit properties

We know that $\lim_{x
ightarrow\infty}\frac{c}{x^{n}} = 0$ for any constant $c$ and positive integer $n$. So, $\lim_{x
ightarrow\infty}\frac{\frac{4}{x}+\frac{4}{x^{2}}+\frac{8}{x^{7}}}{4}=\frac{\lim_{x
ightarrow\infty}\frac{4}{x}+\lim_{x
ightarrow\infty}\frac{4}{x^{2}}+\lim_{x
ightarrow\infty}\frac{8}{x^{7}}}{4}$.

Step3: Calculate the limit

Since $\lim_{x
ightarrow\infty}\frac{4}{x}=0$, $\lim_{x
ightarrow\infty}\frac{4}{x^{2}} = 0$, and $\lim_{x
ightarrow\infty}\frac{8}{x^{7}}=0$, then $\frac{0 + 0+0}{4}=0$.

Answer:

$0$