Question

The value of \[ \lim_{x \to 0} \left[ \frac{1}{x} \ln(1+x) \right] \] is:

• A. \( e \)
• B. 1
• C. 0
• D. \( \frac{1}{e} \)

Hint:

For limits of the form \( \frac{0}{0} \), L'Hopital's Rule can be used to differentiate the numerator and denominator separately and then evaluate the limit.

Answer 1

The Correct Option is B

We are asked to evaluate the limit: \[ \lim_{x \to 0} \left[ \frac{1}{x} \ln(1+x) \right] \] Step 1: Apply L'Hopital's Rule.
The given expression is of the indeterminate form \( \frac{0}{0} \) when \( x \to 0 \), so we can apply L'Hopital's Rule. Differentiating the numerator and denominator separately: Numerator: \[ \frac{d}{dx} \left( \ln(1+x) \right) = \frac{1}{1+x} \] Denominator: \[ \frac{d}{dx} (x) = 1 \] Step 2: Evaluate the limit.
Using L'Hopital's Rule, we get the limit: \[ \lim_{x \to 0} \frac{\frac{1}{1+x}}{1} = \lim_{x \to 0} \frac{1}{1+x} = 1 \] Thus, the value of the limit is 1.