Evaluate `lim_(xto oo)sqrt((x-sinx)/(x+cos^(2)x))`

Options

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

Answer

To evaluate the limit

\[ \lim_{x \to \infty} \sqrt{\frac{x - \sin x}{x + \cos^2 x}}, \]

we can start by analyzing the expression inside the square root.

### Step 1: Rewrite the limit

We can factor out \(x\) from both the numerator and the denominator:

\[ \lim_{x \to \infty} \sqrt{\frac{x(1 - \frac{\sin x}{x})}{x(1 + \frac{\cos^2 x}{x})}}. \]

### Step 2: Simplify the expression

The \(x\) in the numerator and denominator cancels out:

\[ \lim_{x \to \infty} \sqrt{\frac{1 - \frac{\sin x}{x}}{1 + \frac{\cos^2 x}{x}}}. \]

### Step 3: Analyze the behavior of \(\sin x\) and \(\cos^2 x\)

As \(x\) approaches infinity, we know that:

- \(\frac{\sin x}{x}\) approaches 0 because \(\sin x\) is bounded between -1 and 1.
- \(\frac{\cos^2 x}{x}\) also approaches 0 because \(\cos^2 x\) is bounded between 0 and 1.

### Step 4: Substitute the limits

Substituting these limits into our expression gives:

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

### Conclusion

Thus, the limit evaluates to:

\[ \boxed{1}. \]