`lim_(xrarroo)(sqrt(x^(2)+sin^(2)x))/(x+cosx)`

Options

A. is equal to 1
B. is equal to 0
C. is equal to `oo`
D. does not exist

Answer

To solve the limit \( \lim_{x \to \infty} \frac{\sqrt{x^2 + \sin^2 x}}{x + \cos x} \), we can follow these steps:

### Step 1: Simplify the expression
We start by dividing both the numerator and the denominator by \( x \) (the highest power of \( x \) in the denominator):

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

### Step 2: Factor out \( x \) from the square root
In the numerator, we can factor \( x^2 \) out of the square root:

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

### Step 3: Cancel \( x \)
Now, we can cancel \( x \) from the numerator and the denominator (since \( x \) approaches infinity and is positive):

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

### Step 4: Evaluate the limit
As \( x \) approaches infinity, \( \frac{\sin^2 x}{x^2} \) approaches \( 0 \) and \( \frac{\cos x}{x} \) also approaches \( 0 \):

\[ = \frac{\sqrt{1 + 0}}{1 + 0} = \frac{\sqrt{1}}{1} = 1 \]

### Conclusion
Thus, the limit is:

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

### Final Answer
The limit is equal to \( 1 \).

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