Evaluate the following limits :
`Lim_(x to oo) sqrt(((x+sin x)/(x- cos x)))` is equal to

Options

A. 0
B. 1
C. `-1`
D. None of these

Answer

To evaluate the limit \( \lim_{x \to \infty} \sqrt{\frac{x + \sin x}{x - \cos x}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit:
\[ \lim_{x \to \infty} \sqrt{\frac{x + \sin x}{x - \cos x}} \]

### Step 2: Divide numerator and denominator by \( x \)
To simplify the expression, we divide both the numerator and the denominator by \( x \):
\[ = \lim_{x \to \infty} \sqrt{\frac{\frac{x}{x} + \frac{\sin x}{x}}{\frac{x}{x} - \frac{\cos x}{x}}} \]
This simplifies to:
\[ = \lim_{x \to \infty} \sqrt{\frac{1 + \frac{\sin x}{x}}{1 - \frac{\cos x}{x}}} \]

### Step 3: Evaluate the limits of \( \frac{\sin x}{x} \) and \( \frac{\cos x}{x} \)
As \( x \) approaches infinity, both \( \frac{\sin x}{x} \) and \( \frac{\cos x}{x} \) approach 0:
\[ \lim_{x \to \infty} \frac{\sin x}{x} = 0 \quad \text{and} \quad \lim_{x \to \infty} \frac{\cos x}{x} = 0 \]

### Step 4: Substitute the limits into the expression
Substituting these limits back into our expression gives:
\[ = \sqrt{\frac{1 + 0}{1 - 0}} = \sqrt{\frac{1}{1}} = \sqrt{1} \]

### Step 5: Final result
Thus, we find:
\[ \sqrt{1} = 1 \]

Therefore, the limit is:
\[ \lim_{x \to \infty} \sqrt{\frac{x + \sin x}{x - \cos x}} = 1 \]