This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Step 1: Identify the form of the limit. As x , the numerator x^3 approaches , and the denominator e^x also approaches . This is an indeterminate form of type ()/(), so we can apply L'Hôpital's Rule. Step 2: Apply L'Hôpital's Rule for the first time. L'Hôpital's Rule states that if _x c (f(x))/(g(x)) is of the form (0)/(0) or ()/(), then _x c (f(x))/(g(x)) = _x c (f'(x))/(g'(x)). Here, f(x) = x^3 and g(x) = e^x. The derivatives are f'(x) = 3x^2 and g'(x) = e^x. _x (x^3)/(e^x) = _x (3x^2)/(e^x) Step 3: Apply L'Hôpital's Rule for the second time. The limit is still of the form ()/() as x . Let f_1(x) = 3x^2 and g_1(x) = e^x. The derivatives are f_1'(x) = 6x and g_1'(x) = e^x. _x (3x^2)/(e^x) = _x (6x)/(e^x) Step 4: Apply L'Hôpital's Rule for the third time. The limit is still of the form ()/() as x . Let f_2(x) = 6x and g_2(x) = e^x. The derivatives are f_2'(x) = 6 and g_2'(x) = e^x. _x (6x)/(e^x) = _x (6)/(e^x) Step 5: Evaluate the final limit. As x , e^x approaches . Therefore, (6)/(e^x) approaches (6)/(), which is 0. _x (6)/(e^x) = 0 The final answer is 0. What's next?