NIMCET 2026 — Mathematics PYQ

To evaluate this limit as $x\to \infty$, we can simplify the expression by dividing the numerator and denominator by $\sqrt{x}$.

1. Rewrite the expression:

$$\underset{x\to \infty }{\lim }\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}=\underset{x\to \infty }{\lim }\sqrt{\frac{x}{x+\sqrt{x+\sqrt{x}}}}$$

2. Divide inside the square root by $x$:

$$\underset{x\to \infty }{\lim }\sqrt{\frac{1}{1+\frac{\sqrt{x}}{x}+\frac{\sqrt{\sqrt{x}}}{x}}}=\underset{x\to \infty }{\lim }\sqrt{\frac{1}{1+\frac{1}{\sqrt{x}}+\frac{1}{x^{3/4}}}}$$

3. Evaluate the limit:

As $x\to \infty$, terms like $\frac{1}{\sqrt{x}}$ and $\frac{1}{x^{3/4}}$ approach $0$.

$$\sqrt{\frac{1}{1+0+0}}=\sqrt{1}=1$$

Correct Option: (c)