Evaluate
`Lim_(xto 2) (x^(5)-32)/(x^(2)-4)`

Answer

To evaluate the limit

\[ \lim_{x \to 2} \frac{x^5 - 32}{x^2 - 4}, \]

we can follow these steps:

### Step 1: Factor the denominator
The denominator \(x^2 - 4\) can be factored using the difference of squares:

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

### Step 2: Rewrite the limit
Now we can rewrite the limit as:

\[ \lim_{x \to 2} \frac{x^5 - 32}{(x - 2)(x + 2)}. \]

### Step 3: Factor the numerator
Notice that \(32\) can be written as \(2^5\). Therefore, we can apply the formula for the difference of powers:

\[ x^5 - 32 = x^5 - 2^5 = (x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16). \]

### Step 4: Substitute the factored form into the limit
Substituting the factored form of the numerator into the limit gives us:

\[ \lim_{x \to 2} \frac{(x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)}{(x - 2)(x + 2)}. \]

### Step 5: Cancel the common factor
We can cancel the \((x - 2)\) terms in the numerator and denominator:

\[ \lim_{x \to 2} \frac{x^4 + 2x^3 + 4x^2 + 8x + 16}{x + 2}. \]

### Step 6: Substitute \(x = 2\)
Now we can directly substitute \(x = 2\):

\[ \frac{2^4 + 2 \cdot 2^3 + 4 \cdot 2^2 + 8 \cdot 2 + 16}{2 + 2}. \]

Calculating the numerator:

\[ 2^4 = 16, \quad 2 \cdot 2^3 = 16, \quad 4 \cdot 2^2 = 16, \quad 8 \cdot 2 = 16, \quad 16 = 16. \]

Adding these values together:

\[ 16 + 16 + 16 + 16 + 16 = 80. \]

Now, calculating the denominator:

\[ 2 + 2 = 4. \]

### Step 7: Final calculation
Now we can compute the limit:

\[ \frac{80}{4} = 20. \]

### Conclusion
Thus, we conclude that:

\[ \lim_{x \to 2} \frac{x^5 - 32}{x^2 - 4} = 20. \]