NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

$\lim_{x \to \infty} \sqrt{\frac{(x + \sin x)}{(x - \cos x)}}$ is equal to:

Choose the correct answer:

Explanation

Step 1: Write the limit expression

Let the limit be $L$ :

L = \sqrt{\lim_{x \to \infty} \frac{x + \sin x}{x - \cos x}}

Step 2: Divide numerator and denominator by $x$

Inside the square root, we divide every term by $x$ :

\frac{\frac{x}{x} + \frac{\sin x}{x}}{\frac{x}{x} - \frac{\cos x}{x}} = \frac{1 + \frac{\sin x}{x}}{1 - \frac{\cos x}{x}}

Step 3: Apply the limit $x \to \infty$

We know that both $\sin x$ and $\cos x$ are bounded functions, meaning their values always stay between $-1$ and $1$ .

As $x$ approaches infinity, any finite value divided by an infinitely large number tends to zero:

\lim_{x \to \infty} \frac{\sin x}{x} = 0

\lim_{x \to \infty} \frac{\cos x}{x} = 0

Step 4: Calculate the final value

Substitute these zeros back into the expression:

L = \sqrt{\frac{1 + 0}{1 - 0}}

L = \sqrt{\frac{1}{1}}

L = 1

Conclusion:

The value of the limit is $1$ .

Correct Option:

(b) $1$