`lim_(xrarroo) sqrt((x+sinx)/(x-cos x))=`

Options

A. 0
B. 1
C. -1
D. none of these

Answer

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

### Step 1: Rewrite the limit
We start with the expression inside 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 \( x \to \infty \):
- \( \frac{\sin x}{x} \to 0 \)
- \( \frac{\cos x}{x} \to 0 \)

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

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

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