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. Problem 1: Evaluate _x (2x^2 - 7)/(6 - 3x^2) Step 1: Identify the highest power of x in the denominator, which is x^2. Divide every term in the numerator and denominator by x^2. _x (2x^2)/(x^2) - (7)/(x^2)(6)/(x^2) - (3x^2)/(x^2) Step 2: Simplify the expression. _x (2 - 7)/(x^2)(6)/(x^2) - 3 Step 3: Apply the limit property that _x (c)/(x^n) = 0 for any constant c and n > 0. (2 - 0)/(0 - 3) Step 4: Calculate the final value. (2)/(-3) = -(2)/(3) The limit is -(2)/(3). Problem 2: Evaluate _x 3 (x)/(x+2) - (3)/(5)x-3 Step 1: Combine the fractions in the numerator by finding a common denominator. (x)/(x+2) - (3)/(5) = (5x - 3(x+2))/(5(x+2)) Step 2: Simplify the numerator. (5x - 3x - 6)/(5(x+2)) = (2x - 6)/(5(x+2)) Step 3: Substitute the simplified numerator back into the limit expression and rewrite it as a single fraction. _x 3 (2x - 6)/(5(x+2))x-3 = _x 3 (2x - 6)/(5(x+2)(x-3)) Step 4: Factor the numerator to find a common term with the denominator. _x 3 (2(x - 3))/(5(x+2)(x-3)) Step 5: Cancel out the (x-3) term, since x ≠ 3 as x approaches 3. _x 3 (2)/(5(x+2)) Step 6: Substitute x=3 into the simplified expression. (2)/(5(3+2)) = (2)/(5(5)) = (2)/(25) The limit is (2)/(25). Send me the next one 📸