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 are the solutions to the limit problems visible in your image. 1. We need to evaluate the limit: _x 3 (x^3 - 27)/(x^2 - 9) Step 1: Substitute x=3 into the expression. (3^3 - 27)/(3^2 - 9) = (27 - 27)/(9 - 9) = (0)/(0) This is an indeterminate form, so we need to simplify the expression. Step 2: Factor the numerator using the difference of cubes formula a^3 - b^3 = (a-b)(a^2+ab+b^2) and the denominator using the difference of squares formula a^2 - b^2 = (a-b)(a+b). x^3 - 27 = (x-3)(x^2 + 3x + 9) x^2 - 9 = (x-3)(x+3) Step 3: Rewrite the limit with the factored expressions and cancel out the common term (x-3). _x 3 ((x-3)(x^2 + 3x + 9))/((x-3)(x+3)) = _x 3 (x^2 + 3x + 9)/(x+3) Step 4: Substitute x=3 into the simplified expression. (3^2 + 3(3) + 9)/(3+3) = (9 + 9 + 9)/(6) = (27)/(6) Step 5: Simplify the fraction. (27)/(6) = (9)/(2) The limit is (9)/(2). 2. 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. Inside the square root, the highest power is x^4, so sqrt(x^4) = x^2. Step 2: Divide