Step 1: Identify the limit we want to evaluate: lim (x -> 0) (e^x - 1)/x.
Step 2: Check if we can directly substitute x = 0 into the expression. If we do, we get (e^0 - 1)/0 = (1 - 1)/0 = 0/0, which is an indeterminate form.
Step 3: Since we have an indeterminate form, we can use L'Hôpital's Rule. This rule states that if we have 0/0 or ∞/∞, we can take the derivative of the numerator and the derivative of the denominator.
Step 4: Differentiate 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: Differentiate the denominator (x). The derivative of x is 1.
Step 6: Now we can rewrite the limit using the derivatives: lim (x -> 0) (e^x)/(1).
Step 7: Substitute x = 0 into the new expression: e^0/1 = 1/1 = 1.
Step 8: Therefore, the limit is 1.
Limits and Continuity – Understanding how to evaluate limits, particularly those that result in indeterminate forms.
L'Hôpital's Rule – Applying L'Hôpital's Rule to resolve limits that yield 0/0 or ∞/∞ forms by differentiating the numerator and denominator.
Exponential Functions – Recognizing the behavior of the exponential function e^x as x approaches 0.
