Question

Evaluate: $\lim_{x \to 0} \frac{e^{3x}+x^2}{3x-5x^2}$}

• A. $0$
• B. $\infty$
• C. $-1/3$
• D. $-1/5$

Hint:

If denominator → 0 but numerator → constant, limit → infinity.

Answer 1

The Correct Option is B

Concept: Use series expansion.

Step 1: Expand numerator \[ e^{3x} \approx 1 + 3x + \frac{9x^2}{2} \] \[ e^{3x} + x^2 \approx 1 + 3x + \frac{11x^2}{2} \]

Step 2: Denominator \[ 3x - 5x^2 \]

Step 3: Leading terms \[ \frac{1 + 3x}{3x} \rightarrow \infty \] \[ \therefore \text{Limit } = \infty \]