Question

find the limit of the rational function a. as x→∞ and b. as x→ - ∞. write ∞ or - ∞ where appropriate. h(x)=\frac{3x^{4}}{20x^{4}+8x^{3}+2x^{2}} a. lim_{x→∞}\frac{3x^{4}}{20x^{4}+8x^{3}+2x^{2}} = (simplify your answer.)

Explanation:

Step1: Divide numerator and denominator by highest - power term

Divide both the numerator and denominator of the function $\frac{3x^{4}}{20x^{4}+8x^{3}+2x^{2}}$ by $x^{4}$. We get $\lim_{x
ightarrow\infty}\frac{3x^{4}/x^{4}}{(20x^{4}+8x^{3}+2x^{2})/x^{4}}=\lim_{x
ightarrow\infty}\frac{3}{20 + 8/x+2/x^{2}}$.

Step2: Evaluate the limit of each term

As $x
ightarrow\infty$, $\lim_{x
ightarrow\infty}\frac{8}{x}=0$ and $\lim_{x
ightarrow\infty}\frac{2}{x^{2}} = 0$. So, $\lim_{x
ightarrow\infty}\frac{3}{20 + 8/x+2/x^{2}}=\frac{3}{20+0 + 0}$.

Step3: Simplify the result

$\frac{3}{20+0 + 0}=\frac{3}{20}$.

Answer:

$\frac{3}{20}$