Question

find the limit. (if the limit is infinite, enter \\( \infty \\) or \\( -\infty \\), as appropriate. if the limit does not otherwise exist, ent
\\ \lim _{x \
ightarrow-\infty} \frac{3 x^{5}-x}{x^{4}+4} \\

Explanation:

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

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

Step2: Evaluate the limit

As \(x\to-\infty\), \(\frac{1}{x^3}\to0\) and \(\frac{4}{x^4}\to0\). So the limit becomes \(\lim_{x\to -\infty}(3x)\)
Since \(x\to-\infty\), \(3x\to-\infty\)

Answer:

\(-\infty\)