First slide

Introduction to limits

Question

$lim_{x \to \infty} (\sqrt{x^{2} + 2 x - 1} - x) =$

Moderate

Solution

We have,
$\begin{array}{l} lim_{x \to \infty} (\sqrt{x^{2} + 2 x - 1} - x) \\ = lim_{x \to \infty} \frac{x^{2} + 2 x - 1 - x^{2}}{\sqrt{x^{2} + 2 x - 1} + x} \\ = lim_{x \to \infty} \frac{2 x - 1}{\sqrt{x^{2} + 2 x - 1} + x} \\ = lim_{x \to \infty} \frac{2 - \frac{1}{x}}{\sqrt{1 + \frac{2}{x} - \frac{1}{x^{2}}} + 1} = \frac{2}{1 + 1} = 1 \end{array}$

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