Evaluate the following limits :
`Lim _( x to oo) sqrt(x)(sqrt(x+c)-sqrt(x))`

Answer

To evaluate the limit

\[ \lim_{x \to \infty} \sqrt{x} \left( \sqrt{x+c} - \sqrt{x} \right), \]

we will follow these steps:

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

\[ \lim_{x \to \infty} \sqrt{x} \left( \sqrt{x+c} - \sqrt{x} \right). \]

### Step 2: Multiply and divide by the conjugate
To simplify the expression, we can multiply and divide by the conjugate of the expression inside the limit:

\[ \sqrt{x+c} - \sqrt{x} = \frac{(\sqrt{x+c} - \sqrt{x})(\sqrt{x+c} + \sqrt{x})}{\sqrt{x+c} + \sqrt{x}} = \frac{(x+c) - x}{\sqrt{x+c} + \sqrt{x}} = \frac{c}{\sqrt{x+c} + \sqrt{x}}. \]

Thus, we can rewrite the limit as:

\[ \lim_{x \to \infty} \sqrt{x} \cdot \frac{c}{\sqrt{x+c} + \sqrt{x}}. \]

### Step 3: Substitute the expression into the limit
Now substituting back, we have:

\[ \lim_{x \to \infty} \frac{c \sqrt{x}}{\sqrt{x+c} + \sqrt{x}}. \]

### Step 4: Factor out \(\sqrt{x}\)
Next, we factor \(\sqrt{x}\) out of the denominator:

\[ \sqrt{x+c} = \sqrt{x(1 + \frac{c}{x})} = \sqrt{x} \sqrt{1 + \frac{c}{x}}. \]

So the limit becomes:

\[ \lim_{x \to \infty} \frac{c \sqrt{x}}{\sqrt{x} \left( \sqrt{1 + \frac{c}{x}} + 1 \right)} = \lim_{x \to \infty} \frac{c}{\sqrt{1 + \frac{c}{x}} + 1}. \]

### Step 5: Evaluate the limit
As \(x\) approaches infinity, \(\frac{c}{x}\) approaches 0. Therefore:

\[ \sqrt{1 + \frac{c}{x}} \to \sqrt{1 + 0} = 1. \]

Thus, we have:

\[ \lim_{x \to \infty} \frac{c}{1 + 1} = \frac{c}{2}. \]

### Final Answer
The limit evaluates to:

\[ \boxed{\frac{c}{2}}. \]