Question

find the limit of the rational function (a) as x→∞ and (b) as
f(x) = \frac{6x^{4}+4}{x^{4}-x^{2}+x + 7}
a. lim\limits_{x→∞}(\frac{6x^{4}+4}{x^{4}-x^{2}+x + 7}) = 6 (simplify your answer.)
b. lim\limits_{x→ - ∞}(\frac{6x^{4}+4}{x^{4}-x^{2}+x + 7}) = (simplify your answer.)

Explanation:

Step1: Divide by highest - power term

Divide both the numerator and denominator by $x^{4}$. So, $\frac{6x^{4}+4}{x^{4}-x^{2}+x + 7}=\frac{6+\frac{4}{x^{4}}}{1-\frac{1}{x^{2}}+\frac{1}{x^{3}}+\frac{7}{x^{4}}}$.

Step2: Evaluate limit as $x\to-\infty$

As $x\to-\infty$, $\frac{4}{x^{4}}\to0$, $\frac{1}{x^{2}}\to0$, $\frac{1}{x^{3}}\to0$, and $\frac{7}{x^{4}}\to0$.

Answer:

$6$