Calculate the limit: lim (x -> 0) (e^x - 1)/x

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Question: Calculate the limit: lim (x -> 0) (e^x - 1)/x

Options:

Correct Answer: 1

Solution:

Using the definition of the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = e^0 = 1.

Calculate the limit: lim (x -> 0) (e^x - 1)/x

Calculate the limit: lim (x -> 0) (e^x - 1)/x

Step 1: Understand the limit we want to calculate: lim (x -> 0) (e^x - 1)/x.
Step 2: Recognize that this limit can be interpreted as the derivative of the function e^x at the point x = 0.
Step 3: Recall the definition of the derivative: f'(a) = lim (x -> a) (f(x) - f(a))/(x - a).
Step 4: In our case, f(x) = e^x and a = 0, so we have f(0) = e^0 = 1.
Step 5: Substitute into the derivative formula: f'(0) = lim (x -> 0) (e^x - 1)/(x - 0).
Step 6: This simplifies to lim (x -> 0) (e^x - 1)/x, which is what we want to calculate.
Step 7: Knowing that the derivative of e^x is e^x itself, we find that f'(0) = e^0 = 1.
Step 8: Therefore, we conclude that lim (x -> 0) (e^x - 1)/x = 1.

Limit of a Function – Understanding how to evaluate the limit of a function as it approaches a specific point, particularly using derivatives.
Derivative Definition – Recognizing that the limit presented is equivalent to the definition of the derivative of the exponential function at a point.