If `lim_(x to a) (xsqrt(x) - a sqrt(a))/(x - 1) = lim_(x to 3) (x^(3) - 27)/(x - 3)`, find the value of a.

Answer

To solve the problem, we need to evaluate the limit on both sides of the equation and find the value of \( a \).

### Step 1: Evaluate the right-hand side limit

The right-hand side of the equation is:

\[ \lim_{x \to 3} \frac{x^3 - 27}{x - 3} \]

Using the formula for limits, we recognize that \( x^3 - 27 \) can be factored as \( (x - 3)(x^2 + 3x + 9) \). Therefore, we can rewrite the limit as:

\[ \lim_{x \to 3} \frac{(x - 3)(x^2 + 3x + 9)}{x - 3} \]

Since \( x \to 3 \) and \( x - 3 \) cancels out, we have:

\[ \lim_{x \to 3} (x^2 + 3x + 9) \]

Now, substituting \( x = 3 \):

\[ 3^2 + 3(3) + 9 = 9 + 9 + 9 = 27 \]

Thus, the right-hand side limit evaluates to:

\[ \lim_{x \to 3} \frac{x^3 - 27}{x - 3} = 27 \]

### Step 2: Evaluate the left-hand side limit

Now, we evaluate the left-hand side limit:

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

We can apply L'Hôpital's rule here since both the numerator and denominator approach 0 as \( x \to a \). First, we differentiate the numerator and the denominator:

1. The derivative of the numerator \( x\sqrt{x} - a\sqrt{a} \) is:

\[ \frac{d}{dx}(x\sqrt{x}) = \frac{d}{dx}(x^{3/2}) = \frac{3}{2}x^{1/2} \]

2. The derivative of the denominator \( x - 1 \) is:

\[ \frac{d}{dx}(x - 1) = 1 \]

Applying L'Hôpital's rule gives us:

\[ \lim_{x \to a} \frac{\frac{3}{2}x^{1/2}}{1} = \frac{3}{2}a^{1/2} \]

### Step 3: Set the limits equal to each other

Now we set the left-hand side limit equal to the right-hand side limit:

\[ \frac{3}{2}a^{1/2} = 27 \]

### Step 4: Solve for \( a \)

To solve for \( a \), we first multiply both sides by \( \frac{2}{3} \):

\[ a^{1/2} = \frac{27 \cdot 2}{3} = 18 \]

Now, squaring both sides gives:

\[ a = 18^2 = 324 \]

### Final Answer

Thus, the value of \( a \) is:

\[ \boxed{324} \]