Evaluate the limit: lim (x -> 0) [(e^(3x) - 1) / x].
Answer and explanation
Correct answer: A. 3
- Step 1: Recognize the definition of derivative for e^(3x) at x = 0, or apply L'Hopital's rule.
- Step 2: Differentiate numerator: d/dx(e^(3x) - 1) = 3e^(3x).
- Step 3: Differentiate denominator: d/dx(x) = 1.
- Step 4: Evaluate limit: lim (x -> 0) [3e^(3x) / 1] = 3e^0 = 3(1) = 3.
Answer
3.
Why not the others: '1' assumes the standard limit of (e^x - 1)/x without the multiplier 3; '0' evaluates the indeterminate numerator directly; 'e^3' fails to evaluate the derivative at x = 0.
Practise more Calculus and Mathematical Analysis.
Free questions with an explanation for every one, and full timed General Education Diploma (GED Oman) mock tests.
More Calculus and Mathematical Analysis questions
- Evaluate the trigonometric limit: lim (x -> 0) [sin(7x) / (2x)].
- Find the derivative of the rational function f(x) = (3x - 5) / (2x + 1) for x != -1/2.
- What is the slope of the tangent line to the curve y = 2x^3 - 5x^2 + 4 at x = 2?
- Determine the indefinite integral: integral of (6x^2 - 4x + 5) dx.
- Evaluate the definite integral: integral from 0 to 3 of (4x + 1) dx.
- Find the local maximum of the polynomial function f(x) = -x^3 + 3x + 2.