Evaluate the following limits :
`Lim_(x to 2) (x^(10) - 1024)/(x^(5) - 32)`

Answer

To evaluate the limit

\[ \lim_{x \to 2} \frac{x^{10} - 1024}{x^{5} - 32}, \]

we start by substituting \(x = 2\) into the expression.

1. **Substituting \(x = 2\)**:
\[ x^{10} = 2^{10} = 1024 \quad \text{and} \quad x^{5} = 2^{5} = 32. \]
Therefore, substituting \(x = 2\) gives us:
\[ \frac{1024 - 1024}{32 - 32} = \frac{0}{0}. \]
This is an indeterminate form, so we need to simplify the expression.

2. **Factoring the numerator**:
The numerator \(x^{10} - 1024\) can be factored using the difference of squares:
\[ x^{10} - 1024 = (x^5)^2 - (32)^2 = (x^5 - 32)(x^5 + 32). \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 2} \frac{(x^5 - 32)(x^5 + 32)}{x^5 - 32}. \]

3. **Canceling the common terms**:
Since \(x^5 - 32\) is in both the numerator and the denominator, we can cancel it (as long as \(x \neq 2\)):
\[ \lim_{x \to 2} (x^5 + 32). \]

4. **Substituting \(x = 2\) again**:
Now we can substitute \(x = 2\) into the simplified expression:
\[ 2^5 + 32 = 32 + 32 = 64. \]

Thus, the limit is

\[ \boxed{64}. \]