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
Step 1: Introduce a substitution to simplify the limit expression. Let t = (1)/(x-1). As x approaches 1 from the left side (x 1^-), the term x-1 approaches 0 from the negative side (x-1 0^-). Therefore, t = (1)/(x-1) approaches -. So, the limit variable changes from x 1^- to t -. Step 2: Express x-1 in terms of t. From the substitution t = (1)/(x-1), we can write x-1 = (1)/(t). Step 3: Substitute t into the original limit expression. _t - (e^t)/(1)t Step 4: Simplify the expression. _t - t e^t Step 5: Evaluate the simplified limit. This is an indeterminate form of type (-) · 0. To evaluate it, we can rewrite the expression as a fraction. Let u = -t. As t -, u . The limit becomes: _u (-u) e^-u = _u -(u)/(e^u) This is an indeterminate form of type ()/(). We can apply L'Hôpital's Rule. _u (d)/(du)(u)(d)/(du)(e^u) = - _u (1)/(e^u) As u , e^u approaches . Therefore, (1)/(e^u) approaches 0. _u (1)/(e^u) = -0 = 0 The final answer is 0. Send me the next one 📸