Evaluate the following limits :
`Lim_(x to a) (xsqrt(x)-asqrt(a))/(x-a)`

Answer

To evaluate the limit

\[ \lim_{x \to a} \frac{x\sqrt{x} - a\sqrt{a}}{x - a}, \]

we can follow these steps:

### Step 1: Rewrite the expression

We start with the limit:

\[ \lim_{x \to a} \frac{x\sqrt{x} - a\sqrt{a}}{x - a}. \]

### Step 2: Substitute \( x = a + h \)

Let \( h = x - a \). As \( x \to a \), \( h \to 0 \). Thus, we can rewrite the limit as:

\[ \lim_{h \to 0} \frac{(a + h)\sqrt{a + h} - a\sqrt{a}}{h}. \]

### Step 3: Expand \( \sqrt{a + h} \) using Taylor series

Using the binomial expansion for small \( h \):

\[ \sqrt{a + h} \approx \sqrt{a} + \frac{1}{2\sqrt{a}}h. \]

Substituting this back into our limit gives:

\[ \lim_{h \to 0} \frac{(a + h)\left(\sqrt{a} + \frac{1}{2\sqrt{a}}h\right) - a\sqrt{a}}{h}. \]

### Step 4: Simplify the expression

Expanding the numerator:

\[ = \lim_{h \to 0} \frac{a\sqrt{a} + h\sqrt{a} + \frac{1}{2\sqrt{a}}ah + \frac{1}{2\sqrt{a}}h^2 - a\sqrt{a}}{h}. \]

This simplifies to:

\[ = \lim_{h \to 0} \frac{h\sqrt{a} + \frac{1}{2\sqrt{a}}ah + \frac{1}{2\sqrt{a}}h^2}{h}. \]

### Step 5: Factor out \( h \)

Factoring \( h \) out of the numerator:

\[ = \lim_{h \to 0} \left(\sqrt{a} + \frac{1}{2\sqrt{a}}a + \frac{1}{2\sqrt{a}}h\right). \]

### Step 6: Evaluate the limit

As \( h \to 0 \):

\[ = \sqrt{a} + \frac{1}{2\sqrt{a}}a = \sqrt{a} + \frac{a}{2\sqrt{a}} = \sqrt{a} + \frac{1}{2}\sqrt{a} = \frac{3}{2}\sqrt{a}. \]

Thus, the limit evaluates to:

\[ \frac{3}{2}\sqrt{a}. \]

### Final Answer

\[ \lim_{x \to a} \frac{x\sqrt{x} - a\sqrt{a}}{x - a} = \frac{3}{2}\sqrt{a}. \]