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.
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here are the solutions for the remaining limits: Problem 12: Step 1: Substitute x=-1 into the expression. _x -1 (x^2 - x - 2)/(x - 2) = ((-1)^2 - (-1) - 2)/(-1 - 2) Step 2: Simplify the numerator and denominator. (1 + 1 - 2)/(-3) = (0)/(-3) Step 3: Perform the division. (0)/(-3) = 0 The limit is 0. Problem 13: Step 1: Direct substitution of x=1 results in (0)/(0), an indeterminate form. Factor the numerator. x^2 - 1 = (x - 1)(x + 1) Step 2: Simplify the expression by canceling the common factor (x - 1). _x 1 ((x - 1)(x + 1))/(x - 1) = _x 1 (x + 1) Step 3: Substitute x=1 into the simplified expression. 1 + 1 = 2 The limit is 2. Problem 14: Step 1: Direct substitution of x=-5 results in (0)/(0), an indeterminate form. Factor the numerator. x^2 + 3x - 10 = (x + 5)(x - 2) Step 2: Simplify the expression by canceling the common factor (x + 5). _x -5 ((x + 5)(x - 2))/(x + 5) = _x -5 (x - 2) Step 3: Substitute x=-5 into the simplified expression. -5 - 2 = -7 The limit is -7. Problem 15: Step 1: Direct substitution of x=2 results in (0)/(0), an indeterminate form. Factor the denominator. x^2 + x - 6 = (x + 3)(x - 2) Step 2: Simplify the expression by canceling the common factor (x - 2). _x 2 (x - 2)/((x + 3)(x - 2)) = _x 2 (1)/(x + 3) Step 3: Substitute x=2 into the simplified expression. (1)/(2 + 3) = (1)/(5) The limit is (1)/(5). Problem 16: Step 1: Direct substitution of x=1 results in (0)/(0), an indeterminate form. Factor the numerator and denominator. x^2 - 2x + 1 = (x - 1)^2 x^3 - 2x + 1 = (x - 1)(x^2 + x - 1) Step 2: Simplify the expression by canceling the common factor (x - 1). _x 1 ((x - 1)^2)/((x - 1)(x^2 + x - 1)) = _x 1 (x - 1)/(x^2 + x - 1) Step 3: Substitute x=1 into the simplified expression. (1 - 1)/(1^2 + 1 - 1) = (0)/(1) = 0 The limit is 0. Problem 17: Step 1: Divide the numerator and denominator by the highest power of x in the denominator, which is x. _x (2x + 3)/(5x + 7) = _x (2x)/(x) + (3)/(x)(5x)/(x) + (7)/(x) = _x (2 + 3)/(x)5 + (7)/(x) Step 2: As x , terms of the form (c)/(x^n) approach 0. (2 + 0)/(5 + 0) = (2)/(5) The limit is (2)/(5). Problem 18: Step 1: Divide the numerator and denominator by the highest power of x in the denominator, which is x^2. _x (2x^2 + 3)/(5x^2 + 7) = _x (2x^2)/(x^2) + (3)/(x^2)(5x^2)/(x^2) + (7)/(x^2) = _x (2 + 3)/(x^2)5 + (7)/(x^2) Step 2: As x , terms of the form (c)/(x^n) approach 0. (2 + 0)/(5 + 0) = (2)/(5) The limit is (2)/(5). Problem 19: Step 1: Divide the numerator and denominator by the highest power of x in the denominator, which is x^3. _x (x^2 - 4x + 8)/(3x^3) = _x (x^2)/(x^3) - (4x)/(x^3) + (8)/(x^3)(3x^3)/(x^3) = _x (1)/(x) - (4)/(x^2) + (8)/(x^3)3 Step 2: As x , terms of the form (c)/(x^n) approach 0. (0 - 0 + 0)/(3) = (0)/(3) = 0 The limit is 0. Problem 20: Step 1: As x , the denominator x^3 - 7x + 1 approaches . _x (1)/(x^3 - 7x + 1) Step 2: A constant divided by an infinitely large number approaches 0. (1)/() = 0 The limit is 0. Problem 21: Step 1: Divide the numerator and denominator by the highest power of x in the denominator, which is x. _x (x^2 - 7x)/(x + 1) = _x (x^2)/(x) - (7x)/(x)(x)/(x) + (1)/(x) = _x (x - 7)/(1 + 1)x Step 2: As $ ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._