Step 1: Identify the limit we want to find: lim (x -> 0) (x^3)/(e^x - 1).
Step 2: Substitute x = 0 into the expression. We get (0^3)/(e^0 - 1) = 0/0, which is an indeterminate form.
Step 3: Since we have an indeterminate form (0/0), we can use L'Hôpital's Rule. This rule states that we can take the derivative of the numerator and the derivative of the denominator.
Step 4: Differentiate the numerator: The derivative of x^3 is 3x^2.
Step 5: Differentiate the denominator: The derivative of e^x - 1 is e^x.
Step 6: Rewrite the limit using the derivatives: lim (x -> 0) (3x^2)/(e^x).
Step 7: Substitute x = 0 again: We get (3*0^2)/(e^0) = 0/1 = 0.
Step 8: Since we still have an indeterminate form (0/0), we apply L'Hôpital's Rule again.
Step 9: Differentiate the numerator again: The derivative of 3x^2 is 6x.
Step 10: Differentiate the denominator again: The derivative of e^x is e^x.
Step 17: Substitute x = 0: We get 6/(e^0) = 6/1 = 6.
Step 18: Therefore, the limit is 6.
Limits and Indeterminate Forms – The question tests the understanding of limits, particularly how to handle indeterminate forms like 0/0 using L'Hôpital's Rule.
L'Hôpital's Rule – The application of L'Hôpital's Rule is crucial for solving limits that result in indeterminate forms.
Exponential Functions – Understanding the behavior of the exponential function e^x as x approaches 0 is essential for evaluating the limit.
