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

Answer

To evaluate the limit

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

we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression inside the limit:

\[ \sqrt{\frac{x - \sin x}{x + \cos^2 x}}. \]

### Step 2: Factor out \(x\) from the numerator and denominator
We can factor \(x\) out of both the numerator and the denominator:

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

### Step 3: Simplify the expression
The \(x\) in the numerator and denominator cancels out:

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

### Step 4: Evaluate the limit as \(x\) approaches infinity
Now, we need to evaluate the limit of the expression as \(x\) approaches infinity:

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

As \(x\) approaches infinity, \(\frac{\sin x}{x}\) approaches \(0\) (since \(\sin x\) is bounded between -1 and 1), and \(\frac{\cos^2 x}{x}\) also approaches \(0\) (since \(\cos^2 x\) is also bounded between 0 and 1). Therefore, we have:

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

### Final Answer
Thus, the limit is:

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