Evaluate the following limits :
`Lim_(x to 2) ((x^(8)-16)/(x^(4)-4) +(x^(2)-9)/(x-3))`

Answer

To evaluate the limit

\[ \lim_{x \to 2} \left( \frac{x^8 - 16}{x^4 - 4} + \frac{x^2 - 9}{x - 3} \right), \]

we will simplify each term separately.

### Step 1: Simplify the first term \(\frac{x^8 - 16}{x^4 - 4}\)

The expression \(x^8 - 16\) can be factored using the difference of squares:

\[ x^8 - 16 = (x^4 - 4)(x^4 + 4). \]

Thus, we can rewrite the first term as:

\[ \frac{x^8 - 16}{x^4 - 4} = \frac{(x^4 - 4)(x^4 + 4)}{x^4 - 4}. \]

For \(x \neq 2\), we can cancel \(x^4 - 4\):

\[ \frac{x^8 - 16}{x^4 - 4} = x^4 + 4. \]

### Step 2: Simplify the second term \(\frac{x^2 - 9}{x - 3}\)

The expression \(x^2 - 9\) can also be factored using the difference of squares:

\[ x^2 - 9 = (x - 3)(x + 3). \]

Thus, we can rewrite the second term as:

\[ \frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3}. \]

For \(x \neq 3\), we can cancel \(x - 3\):

\[ \frac{x^2 - 9}{x - 3} = x + 3. \]

### Step 3: Combine the simplified terms

Now, we can combine the simplified terms:

\[ \lim_{x \to 2} \left( x^4 + 4 + x + 3 \right). \]

### Step 4: Substitute \(x = 2\)

Now we substitute \(x = 2\):

\[ = 2^4 + 4 + 2 + 3. \]

Calculating each term:

\[ = 16 + 4 + 2 + 3 = 25. \]

### Final Answer

Thus, the limit is

\[ \boxed{25}. \]

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