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

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

Step 1: Understand the limit we want to find: lim (x -> 0) (e^x - 1)/x.
Step 2: Recognize that as x approaches 0, both the numerator (e^x - 1) and the denominator (x) approach 0. This is an indeterminate form (0/0).
Step 3: To resolve this, we can use L'Hôpital's Rule, which states that if we have an indeterminate form, we can take the derivative of the numerator and the derivative of the denominator.
Step 4: Find the derivative of the numerator e^x - 1. The derivative of e^x is e^x, and the derivative of -1 is 0. So, the derivative of the numerator is e^x.
Step 5: Find the derivative of the denominator x. The derivative of x is 1.
Step 6: Now apply L'Hôpital's Rule: lim (x -> 0) (e^x - 1)/x = lim (x -> 0) e^x/1.
Step 7: Evaluate the limit: as x approaches 0, e^x approaches e^0, which is 1.
Step 8: Therefore, 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 Interpretation – Using the definition of the derivative to find limits, particularly for exponential functions.