Evaluate the limit: lim (x -> 0) [(e^(3x) - 1) / x].
Answer and explanation
Correct answer: A. 3
- Step 1: This is an indeterminate form 0/0. Apply L'Hôpital's Rule or the definition of the derivative.
- Step 2: Differentiate numerator and denominator with respect to x: d/dx [e^(3x) - 1] = 3 * e^(3x), and d/dx [x] = 1.
- Step 3: Evaluate the limit: lim (x -> 0) [3 e^(3x) / 1] = 3 e^0 = 3 * 1 = 3.
Answer
3.
Why not the others: '1' forgets the chain rule derivative of the exponent 3x; '0' assumes direct evaluation; 'e^3' fails to evaluate the limit as x approaches 0.
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