Evaluate `lim_(x to a) (sqrt(x) + sqrt(a))/(x + a)`

Answer

To evaluate the limit \( \lim_{x \to a} \frac{\sqrt{x} + \sqrt{a}}{x + a} \), we can follow these steps:

### Step 1: Substitute \( x = a \)
We start by substituting \( x \) with \( a \) in the expression:

\[ \frac{\sqrt{a} + \sqrt{a}}{a + a} \]

### Step 2: Simplify the expression
Now, simplify the numerator and denominator:

\[ \frac{2\sqrt{a}}{2a} \]

### Step 3: Cancel common factors
We can cancel the common factor of 2 in the numerator and denominator:

\[ \frac{\sqrt{a}}{a} \]

### Step 4: Further simplify
This can be rewritten as:

\[ \frac{\sqrt{a}}{a} = \frac{1}{\sqrt{a}} \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to a} \frac{\sqrt{x} + \sqrt{a}}{x + a} = \frac{1}{\sqrt{a}} \]