Evaluate each limit. 


lim x→0 e^4x−1 / e^x−1

Verified step by step guidance

1. Step 1: Recognize that the limit is in an indeterminate form 0/0 as x approaches 0. This suggests that L'Hôpital's Rule might be applicable.
2. Step 2: Apply L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
3. Step 3: Differentiate the numerator e^{4x} - 1 with respect to x. The derivative of e^{4x} is 4e^{4x}, and the derivative of -1 is 0.
4. Step 4: Differentiate the denominator e^x - 1 with respect to x. The derivative of e^x is e^x, and the derivative of -1 is 0.
5. Step 5: Substitute the derivatives back into the limit expression and evaluate the new limit: lim x→0 (4e^{4x}) / (e^x). Simplify the expression and evaluate the limit as x approaches 0.