Question

Evaluate $\underset{x\to 2}{lim}\frac{x^{10}-1024}{x^5-32}$

• A. 16
• B. 32
• C. 64
• D. 128

Answer 1

The correct option is C

64

We have, $\underset{x\to 2}{lim}\frac{x^{10}-1024}{x^5-32}$

At x=2, the expression $\frac{x^{10}-1024}{x^5-32}$ is the form $\frac{0}{0}$.It is an indeterminant form. Already, we have solved these types of problems using standard limits.

Now, we will solve it with a ver effective method called L'hospital rule.

here, if we have indeterminate form of $\frac{0}{0}$ . so, we need to do is differentiate the numerator and differentaiate the denominator and take the limit.

= $\underset{x\to 2}{lim}\frac{\frac{d}{dx}(x^{10}-1024)}{\frac{d}{dx}(x^5-32)}$

= $\underset{x\to 2}{lim}\frac{10x^9-0}{5x^4-0}$ $\{\frac{d}{dx}{x}^n=n{x}^{n-1}\}$

= $\underset{x\to 2}{lim}2x^5=2\times 2^5=64$