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Question

${lim}_{x \to 0} \frac{\sqrt{1 + x^{2}} - \sqrt{1 + x}}{x}$

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Solution

Verified by Toppr

The expression takes the form $\frac{0}{0}$
Using L`Hospitals Rule
${lim}_{x \to 0} \frac{2 x}{2 \sqrt{1 + x^{2}}} - \frac{1}{2 \sqrt{1 + x}} \frac{2 (0)}{2 \sqrt{1 + 0}} - \frac{1}{2 \sqrt{1 + 0}} = - \frac{1}{2}$

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{lim}_{x \to 0} \frac{\sqrt{1 + x^{2}} - \sqrt{1 + x}}{x}

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