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
Here is the solution to the limit problem. We need to evaluate the limit: _x (4x^2 - 5x + 1)/(sqrt(9x^4 + 2x^2 - 6)) Step 1: Identify the highest power of x in the denominator. The highest power of x inside the square root is x^4. When x , sqrt(x^4) = x^2. Therefore, we will divide every term in the numerator and the denominator by x^2. For terms inside the square root, we divide by x^4. Step 2: Divide each term by the identified highest power of x. _x (4x^2)/(x^2) - (5x)/(x^2) + (1)/(x^2)sqrt((9x^4)/(x^4) + (2x^2)/(x^4) - (6)/(x^4)) Step 3: Simplify the terms. _x (4 - 5)/(x) + (1)/(x^2)sqrt(9 + (2)/(x^2) - (6)/(x^4)) Step 4: Apply the limit as x . As x , any term of the form (c)/(x^n) (where c is a constant and n > 0) approaches 0. (4 - 0 + 0)/(sqrt(9 + 0 - 0)) Step 5: Simplify the expression. (4)/(sqrt(9)) = (4)/(3) The limit is (4)/(3). 3 done, 2 left today. You're making progress.