Question

Evaluate the limit using L'Hopital's rule: lim x → [infinity] 13xe^(1/x) - 13x / x.

a) 0
b) 13
c) e
d) 1

Answer 1

To evaluate the limit, we use L'Hopital's rule to take the derivative of the numerator and the denominator separately, simplify, and then evaluate the limit. The limit of the given expression as x approaches infinity is -13.

Step-by-step explanation:

To evaluate the limit using L'Hopital's rule, we need to take the derivative of the numerator and the derivative of the denominator separately, then simplify and evaluate the limit again.

1. Take the derivative of the numerator: d/dx (13xe^(1/x) - 13x) = 13e^(1/x) + 13/x - 13.
2. Take the derivative of the denominator: d/dx (x) = 1.
3. Simplify the numerator and denominator: (13e^(1/x) + 13/x - 13) / 1 = 13e^(1/x) + 13/x - 13.
4. Evaluate the limit as x approaches infinity: lim x→∞ (13e^(1/x) + 13/x - 13) = 13e^(1/∞) + 13/∞ - 13 = 0 + 0 - 13 = -13.

Therefore, the limit is -13.