First slide

Introduction to limits

Question

The value of $lim_{x \to 1} \frac{\sqrt[3]{7 + x^{2}} - \sqrt{3 + x^{2}}}{x - 1}$ is

Moderate

A

$- \frac{1}{3}$
B

$\frac{3}{2}$
C

$- \frac{1}{4}$
D

$\frac{1}{4}$

Solution

$lim_{x \to 1} \frac{\sqrt[3]{7 + x^{2}} - \sqrt{3 + x^{2}}}{x - 1}$
It is an indeterminate $(\frac{0}{0})$ form
On applying L' Hospital's rule, we get
$\begin{array}{l} lim_{x \to 1} \frac{\frac{1}{3} {(7 + x^{2})}^{\frac{- 2}{3}} \cdot 2 x - \frac{1}{2} {(3 + x^{2})}^{\frac{- 1}{2}} \cdot 2 x}{1 - 0} \\ = \frac{1}{3} (8)^{\frac{- 2}{3}} \cdot 2 - \frac{2}{2} (4)^{\frac{- 1}{2}} = \frac{2}{3} (2)^{- 2} - \frac{2}{2} (2)^{- 1} \\ = \frac{2}{3} [\frac{1}{4}] - \frac{1}{2} = - \frac{1}{3} \end{array}$

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