Q97: Evaluate $\lim_{x \to +\infty} \frac{x}{\sqrt{x^2 + 1000}}$: - Mathematics - ICAR AIEEA PG 2024 - Statistical Sciences | Agrigyan

Evaluating the limit as $x \to +\infty$:
Factor $x^2$ out of the square root in the denominator (for $x \gt 0$, $\sqrt{x^2} = x$):

$$\lim_{x \to +\infty} \frac{x}{\sqrt{x^2\left(1 + \frac{1000}{x^2}\right)}} = \lim_{x \to +\infty} \frac{x}{x\sqrt{1 + \frac{1000}{x^2}}}$$ $$= \lim_{x \to +\infty} \frac{1}{\sqrt{1 + \frac{1000}{x^2}}} = \frac{1}{\sqrt{1 + 0}} = 1$$