Evaluate the following limits: `lim_(xrarr0)((xcosx+sinx))/((x^(2)+tanx))`

Explanation

To evaluate the limit

\[ \lim_{x \to 0} \frac{x \cos x + \sin x}{x^2 + \tan x} \]

we can follow these steps:

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

\[ \lim_{x \to 0} \frac{x \cos x + \sin x}{x^2 + \tan x} \]

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

\[ = \lim_{x \to 0} \frac{x(\cos x + \frac{\sin x}{x})}{x^2(1 + \frac{\tan x}{x^2})} \]

### Step 3: Simplify the limit
This simplifies to:

\[ = \lim_{x \to 0} \frac{\cos x + \frac{\sin x}{x}}{x(1 + \frac{\tan x}{x^2})} \]

### Step 4: Evaluate the limits separately
Now we can evaluate the limits separately:

1. \(\lim_{x \to 0} \cos x = 1\)
2. \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
3. \(\lim_{x \to 0} \frac{\tan x}{x^2} = \lim_{x \to 0} \frac{\sin x}{x^2 \cos x} = \lim_{x \to 0} \frac{1}{x \cos x} = \infty\)

### Step 5: Substitute the limits
Substituting these values back into our expression gives:

\[ = \frac{1 + 1}{0(1 + \infty)} \]

### Step 6: Analyze the limit
Since the denominator approaches \(0\) and the numerator approaches \(2\), the limit approaches:

\[ \frac{2}{0} \to \infty \]

### Conclusion
Thus, the limit is:

\[ \lim_{x \to 0} \frac{x \cos x + \sin x}{x^2 + \tan x} = \infty \]