Evaluate the following limits :
`Lim_(x to 2) (x^(2)(x^(2)-4))/(x-2)`

Answer

To evaluate the limit

\[ \lim_{x \to 2} \frac{x^2 (x^2 - 4)}{x - 2} \]

we can follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = 2 \) into the expression to check if we can directly evaluate the limit.

\[ \frac{2^2 (2^2 - 4)}{2 - 2} = \frac{4(4 - 4)}{0} = \frac{4 \cdot 0}{0} = \frac{0}{0} \]

This is an indeterminate form (0/0), so we need to simplify the expression.

### Step 2: Factor the expression
Notice that \( x^2 - 4 \) can be factored using the difference of squares:

\[ x^2 - 4 = (x - 2)(x + 2) \]

Substituting this back into our limit gives:

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

### Step 3: Cancel the common terms
We can cancel the \( (x - 2) \) in the numerator and denominator:

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

### Step 4: Substitute the limit value again
Now we can substitute \( x = 2 \) into the simplified expression:

\[ = 2^2 (2 + 2) = 4 \cdot 4 = 16 \]

### Final Answer
Thus, the limit is

\[ \boxed{16} \]