find the limit. (if the limit is infinite, enter ∞ or -∞, as appr\n lim _ { x \rightarrow - infty } \frac {…

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

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

Answer

Explanation:

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

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

Step2: Evaluate the limit

As (x\rightarrow-\infty), (\frac{1}{x^{3}}\rightarrow0) and (\frac{4}{x^{4}}\rightarrow0). So (\lim_{x\rightarrow-\infty}\frac{3x-\frac{1}{x^{3}}}{1+\frac{4}{x^{4}}}=\lim_{x\rightarrow-\infty}(3x)) Since (x\rightarrow-\infty), (3x\rightarrow-\infty)

Answer:

(-\infty)