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 -, e^x approaches 0, and x^2 approaches . The limit is of the indeterminate form 0 × . Step 2: Rewrite the expression to apply L'Hôpital's Rule. To use L'Hôpital's Rule, we need the expression in the form (0)/(0) or ()/(). We can rewrite e^x × x^2 as (x^2)/(e^-x). Step 3: Apply L'Hôpital's Rule for the first time. As x -, the numerator x^2 and the denominator e^-x e^-(-) = e^ = . This is an ()/() form. _x - (x^2)/(e^-x) = _x - (d)/(dx)(x^2)(d)/(dx)(e^-x) = _x - (2x)/(-e^-x) Step 4: Apply L'Hôpital's Rule for the second time. As x -, the numerator 2x - and the denominator -e^-x -e^-(-) = -. This is still an ()/() form. _x - (2x)/(-e^-x) = _x - (d)/(dx)(2x)(d)/(dx)(-e^-x) = _x - (2)/(-(-e^-x)) = _x - (2)/(e^-x) Step 5: Evaluate the final limit. As x -, e^-x approaches e^-(-) = e^ = . Therefore, the expression (2)/(e^-x) approaches (2)/(), which is 0. _x - (2)/(e^-x) = 0 The final answer is 0. What's next?