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Question
\[\lim_{x \to \infty} \sqrt{x + 1} - \sqrt{x}\]
Solution
\[\lim_{x \to \infty} \left( \sqrt{x + 1} - \sqrt{x} \right)\]
It is of the form ∞–∞.
On rationalising, we get:
\[\lim_{x \to \infty} \left( \sqrt{x + 1} - \sqrt{x} \right) \times \left( \frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} \right)\]
\[ = \lim_{x \to \infty} \left( \frac{x + 1 - x}{\left( \sqrt{x + 1} + \sqrt{x} \right)} \right)\]
\[ = \frac{1}{\infty}\]
\[ = 0\]
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