Question

Evaluate the limit: \[ \lim_{x \to 0} \frac{\sqrt[3]{8 + x} - 2}{x}. \]

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{12} \)

Hint:

For cube roots, use the binomial expansion for small values of \( x \) to simplify the limit expression.

Answer 1

The Correct Option is D

We are given: \[ L = \lim_{x \to 0} \frac{\sqrt[3]{8 + x} - 2}{x}. \] We apply the binomial expansion for small \( x \): \[ \sqrt[3]{8 + x} \approx 2 + \frac{x}{12} \text{ for small } x. \] Thus, the given expression becomes: \[ \lim_{x \to 0} \frac{\left(2 + \frac{x}{12}\right) - 2}{x} = \lim_{x \to 0} \frac{\frac{x}{12}}{x} = \frac{1}{12}. \]