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Question
\[\lim_{x \to 0} \frac{\sqrt{a^2 + x^2} - a}{x^2}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{\sqrt{a^2 + x^2} - a}{x^2} \right]\]
On putting x = 0 in the expression \[\sqrt{a^2 + x^2} - a\] it becomes \[\frac{0}{0} .\] Rationalising the numerator:
\[\lim_{x \to 0} \left[ \frac{\left( \sqrt{a^2 + x^2} - a \right)\left( \sqrt{a^2 + x^2} + a \right)}{x^2 \left( \sqrt{a^2 + x^2} + a \right)} \right]\]
= \[\lim_{x \to 0} \left[ \frac{a^2 + x^2 - a^2}{x^2 \left( \sqrt{a^2 + x^2} + a \right)} \right]\]
= \[\frac{1}{\sqrt{a^2} + a}\]
= \[\frac{1}{2a}\]
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