Evaluate each limit.

lim x→0 e^4x−1 / e^x−1

검증된 단계별 안내

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.

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.

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.

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.

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.

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주요 개념

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the function as x approaches 0. Understanding limits is crucial for evaluating functions that may not be directly computable at specific points, especially when they lead to indeterminate forms.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = e^(kx), where e is the base of the natural logarithm and k is a constant. These functions are characterized by their rapid growth and unique properties, such as the fact that the derivative of e^x is e^x. In the limit problem, we are dealing with the exponential functions e^(4x) and e^x, which will influence the behavior of the limit as x approaches 0.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to such a form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule is particularly useful in the given limit problem, as both the numerator and denominator approach 0 as x approaches 0.

Evaluate each limit. lim x→0 e^4x−1 / e^x−1