Q. 15 - Calculus

Question

Consider the limit expression

$\underset{x\to 3}{\lim }\frac{\frac{1}{x-3}-\frac{1}{x}}{x-3}$

Calculate the limit.

Step-by-Step Solution

Answer

The limit  of the expression is $\infty$.

Step 1. Given

The Given information is

$\underset{x\to 3}{\lim }\frac{\frac{1}{x-3}-\frac{1}{x}}{x-3}$

Step 2. Calculation

$$\begin{gathered} \underset{x\to 3}{\lim }\frac{\frac{1}{x-3}-\frac{1}{x}}{x-3} \\ \underset{x\to 3}{\lim }\frac{\frac{1}{3-3}-\frac{1}{3}}{3-3} \\ =\underset{x\to 0^{+}}{\lim }\frac{\infty -\infty }{0} \\ =\infty \end{gathered}$$