Q. 16 - Calculus

Question

Consider the limit expression

$\underset{x\to 4}{\lim }\frac{2-\sqrt{x}}{4-x}$

Calculate the limit.

Step-by-Step Solution

Answer

The limit  of the expression is $\infty .$

Step 1. Given

The given expression is 

$\underset{x\to 4}{\lim }\frac{2-\sqrt{x}}{4-x}$

Step 2. Calculation

The evaluated expression is 

$$\begin{gathered} \underset{x\to 4}{\lim }\frac{2-\sqrt{x}}{4-x} \\ =\frac{{\lim }_{x\to 4}(2-\sqrt{x})}{{\lim }_{x\to 4}(4-x)} [\mathrm{Quotient} \mathrm{rule}] \\ =\frac{{\lim }_{x\to 4}\left(2\right)-{\lim }_{x\to 4}\left(\sqrt{x}\right)}{{\lim }_{x\to 4}\left(4\right)-{\lim }_{x\to 4}\left(x\right)} [\mathrm{Difference} \mathrm{Rule}] \\ =\frac{2-\sqrt{4}}{4-4} \\ =\frac{2-2}{0} \\ =\infty \end{gathered}$$