Q. 38 - Calculus

Question

Calculate each limit in Exercises 35-80.

$\underset{x\to \infty }{\lim }\left(-5x^{\frac{3}{5}}\right)$

Step-by-Step Solution

Answer

The limit of the function is $\underset{x\to \infty }{\lim }\left(-5x^{\frac{3}{5}}\right)=-\infty$.

Step 1. Given information.

We have:

$\underset{x\to \infty }{\lim }\left(-5x^{\frac{3}{5}}\right)$

Step 2. Find the limits.

The function is $\underset{x\to \infty }{\lim }\left(-5x^{\frac{3}{5}}\right)$.

$$\begin{gathered} \underset{x\to a}{\lim }\left[c·f\left(x\right)\right]=c·\underset{x\to a}{\lim }f\left(x\right) \\ =-5·\underset{x\to \infty \hspace{0.17em}}{\lim }\left(x^{\frac{3}{5}}\right) \\ \underset{x\to a}{\lim }{\left[f\left(x\right)\right]}^b={\left[\underset{x\to a}{\lim }f\left(x\right)\right]}^b \\ =-5{\left(\underset{x\to \infty \hspace{0.17em}}{\lim }\left(x\right)\right)}^{\frac{3}{5}} \\ \mathrm{Apply}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{common}\hspace{0.17em}\mathrm{limit}:\hspace{0.17em}\underset{x\to \infty \hspace{0.17em}}{\lim }\left(x\right)=\infty \hspace{0.17em} \\ =-5{\infty }^{\frac{3}{5}} \\ =-\infty \hspace{0.17em} \end{gathered}$$