Question

find the limit. (if the limit is infinite, enter ∞ or -∞, as appr
lim _ { x
ightarrow - infty } \frac { 3 x ^ { 5 } - x } { x ^ { 4 } + 4 }

Explanation:

Step1: Divide numerator and denominator by \(x^{4}\)

$$\lim_{x ightarrow-\infty}\frac{3x^{5}-x}{x^{4} + 4}=\lim_{x ightarrow-\infty}\frac{\frac{3x^{5}}{x^{4}}-\frac{x}{x^{4}}}{\frac{x^{4}}{x^{4}}+\frac{4}{x^{4}}}=\lim_{x ightarrow-\infty}\frac{3x-\frac{1}{x^{3}}}{1 + \frac{4}{x^{4}}}$$

Step2: Evaluate the limit

As \(x
ightarrow-\infty\), \(\frac{1}{x^{3}}
ightarrow0\) and \(\frac{4}{x^{4}}
ightarrow0\). So \(\lim_{x
ightarrow-\infty}\frac{3x-\frac{1}{x^{3}}}{1+\frac{4}{x^{4}}}=\lim_{x
ightarrow-\infty}(3x)\)
Since \(x
ightarrow-\infty\), \(3x
ightarrow-\infty\)

Answer:

\(-\infty\)