Q. 36 - Calculus

Question

Calculate each limit in Exercises 35-80.

$\underset{x\to \hspace{0.17em}0}{\lim }\left(2x^{-\frac{3}{4}}\right)$

Step-by-Step Solution

Answer

The function $\underset{x\to \hspace{0.17em}0}{\lim }\left(2x^{-\frac{3}{4}}\right)$ does not exist.

Step 1. Given information.

We have:

$\underset{x\to \hspace{0.17em}0}{\lim }\left(2x^{-\frac{3}{4}}\right)$

Step 2. Find the limits.

The function is $\underset{x\to \hspace{0.17em}0}{\lim }\left(2x^{-\frac{3}{4}}\right)$.

$\mathrm{If}\hspace{0.17em}\underset{x\to a-}{\lim }f\left(x\right)\ne \underset{x\to a+}{\lim }f\left(x\right)\hspace{0.17em}\mathrm{then}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{limit}\hspace{0.17em}\mathrm{does}\hspace{0.17em}\mathrm{not}\hspace{0.17em}\mathrm{exist}$

$$\begin{gathered} \underset{x\to \hspace{0.17em}{0}^{+}}{\lim }\left(2x^{-\frac{3}{4}}\right)=2\underset{x\to \hspace{0.17em}0}{\lim }\left(\frac{1}{x^{\frac{3}{4}}}\right)=\infty \\ \underset{x\to \hspace{0.17em}{0}^{-}}{\lim }\left(2x^{-\frac{3}{4}}\right)=2\underset{x\to \hspace{0.17em}{0}^{-}}{\lim }\left(\frac{1}{x^{\frac{3}{4}}}\right)=\mathrm{Does} \mathrm{not} \mathrm{exist} \end{gathered}$$

Therefore, the limit does not exist.