`lim_(x to infty) sqrt((x + sinx)/(x - cosx))=`

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 can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression inside the square root:

\[ \sqrt{\frac{x + \sin x}{x - \cos x}} = \sqrt{\frac{x(1 + \frac{\sin x}{x})}{x(1 - \frac{\cos x}{x})}}. \]

### Step 2: Simplify the expression
Next, we can simplify this expression by canceling \(x\) in the numerator and denominator:

\[ = \sqrt{\frac{1 + \frac{\sin x}{x}}{1 - \frac{\cos x}{x}}}. \]

### Step 3: Evaluate the limits of \(\frac{\sin x}{x}\) and \(\frac{\cos x}{x}\)
As \(x\) approaches infinity, we know that:

\[ \lim_{x \to \infty} \frac{\sin x}{x} = 0 \quad \text{and} \quad \lim_{x \to \infty} \frac{\cos x}{x} = 0. \]

### Step 4: Substitute the limits into the expression
Substituting these limits into our expression gives:

\[ \lim_{x \to \infty} \sqrt{\frac{1 + 0}{1 - 0}} = \sqrt{\frac{1}{1}} = \sqrt{1} = 1. \]

### Final Result
Thus, we conclude that:

\[ \lim_{x \to \infty} \sqrt{\frac{x + \sin x}{x - \cos x}} = 1. \]