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 for the limits: First Limit: Step 1: Substitute x=3 into the expression (x + 2)(x - 5). _x 3 (x + 2)(x - 5) = (3 + 2)(3 - 5) Step 2: Simplify the terms in the parentheses. (5)(-2) Step 3: Multiply the values. 5 × -2 = -10 The limit is -10. Second Limit: Step 1: Substitute x=1 into the numerator and denominator of the expression (x + 1)/(x + 2). _x 1 (x + 1)/(x + 2) = (1 + 1)/(1 + 2) Step 2: Simplify the numerator and denominator. (2)/(3) The limit is (2)/(3). Third Limit: Step 1: Substitute x=-1 into the numerator and denominator of the expression (x - 1)/(x + 2). _x -1 (x - 1)/(x + 2) = (-1 - 1)/(-1 + 2) Step 2: Simplify the numerator and denominator. (-2)/(1) Step 3: Perform the division. (-2)/(1) = -2 The limit is -2. Fourth Limit: Step 1: Substitute x=3 into the numerator and denominator of the expression (x - 3)/(x^2). _x 3 (x - 3)/(x^2) = (3 - 3)/(3^2) Step 2: Simplify the numerator and denominator. (0)/(9) Step 3: Perform the division. (0)/(9) = 0 The limit is 0. Fifth Limit: Step 1: Substitute x=-1 into the numerator and denominator of the expression (x^2 + 1)/(3x^2 - 2x + 5). _x -1 (x^2 + 1)/(3x^2 - 2x + 5) = ((-1)^2 + 1)/(3(-1)^2 - 2(-1) + 5) Step 2: Simplify the numerator and denominator. (1 + 1)/(3(1) + 2 + 5) = (2)/(3 + 2 + 5) Step 3: Add the terms in the denominator. (2)/(10) Step 4: Simplify the fraction. (2)/(10) = (1)/(5) The limit is (1)/(5). That's 2 down. 3 left today — send the next one.