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: Substitute x = -2 into the expression to check for an indeterminate form. (-2+2)/(sqrt(-2+6)-2) = (0)/(sqrt(4)-2) = (0)/(2-2) = (0)/(0) Since this is an indeterminate form, we need to simplify the expression. Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(x+6)+2. _x -2 (x+2)/(sqrt(x+6)-2) × sqrt(x+6)+2sqrt(x+6)+2 Step 3: Multiply the terms. Use the difference of squares formula (a-b)(a+b) = a^2-b^2 for the denominator. _x -2 (x+2)(sqrt(x+6)+2)(sqrt(x+6))^2 - 2^2 _x -2 (x+2)(sqrt(x+6)+2)x+6-4 _x -2 (x+2)(sqrt(x+6)+2)x+2 Step 4: Cancel out the common factor (x+2) from the numerator and denominator. _x -2 (sqrt(x+6)+2) Step 5: Substitute x = -2 into the simplified expression. sqrt(-2+6)+2 = sqrt(4)+2 = 2+2 = 4 The limit is 4. The final answer is 4. Send me the next one 📸