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
To evaluate the limit, first, substitute y=9 into the expression to check for an indeterminate form. sqrt(9)-39^2-8(9)-9 = (3-3)/(81-72-9) = (0)/(0) Since we have the indeterminate form (0)/(0), we need to simplify the expression. Step 1: Factor the denominator. The denominator is a quadratic expression y^2 - 8y - 9. We look for two numbers that multiply to -9 and add to -8. These numbers are -9 and 1. y^2 - 8y - 9 = (y-9)(y+1) Step 2: Rationalize the numerator. Multiply the numerator and the denominator by the conjugate of the numerator, which is sqrt(y)+3. _y 9 sqrt(y)-3(y-9)(y+1) × sqrt(y)+3sqrt(y)+3 Step 3: Simplify the numerator using the difference of squares formula (a-b)(a+b) = a^2-b^2. (sqrt(y)-3)(sqrt(y)+3) = (sqrt(y))^2 - 3^2 = y-9 Step 4: Substitute the simplified numerator back into the limit expression. _y 9 (y-9)/((y-9)(y+1)(sqrt(y)+3)) Step 5: Cancel out the common factor (y-9) from the numerator and denominator. Since y 9, y ≠ 9, so y-9 ≠ 0. _y 9 (1)/((y+1)(sqrt(y)+3)) Step 6: Substitute y=9 into the simplified expression. (1)/((9+1)(sqrt(9)+3)) = (1)/((10)(3+3)) = (1)/((10)(6)) = (1)/(60) The limit is: (1)/(60) Send me the next one 📸