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
Hey Eniola, good to see you again. Here are the solutions for the limit problems: a) Step 1: Substitute x=2 into the function. _x 2 (8 - 3x + 2x^2) = 8 - 3(2) + 2(2)^2 Step 2: Simplify the expression. = 8 - 6 + 2(4) = 8 - 6 + 8 = 2 + 8 = 10 The limit is 10. b) Step 1: Substitute t=-3 into the function. _t -3 (1 - 2t^3) = 1 - 2(-3)^3 Step 2: Simplify the expression. = 1 - 2(-27) = 1 + 54 = 55 The limit is 55. c) Step 1: Substitute y=0 into the function. _y 0 (3y^4 + 2y^2 - 3y - 24) = 3(0)^4 + 2(0)^2 - 3(0) - 24 Step 2: Simplify the expression. = 0 + 0 - 0 - 24 = -24 The limit is -24. d) Step 1: Substitute t=-1 into the function. _t -1 (3 - 3t)/(5 + 2t) = (3 - 3(-1))/(5 + 2(-1)) Step 2: Simplify the expression. = (3 + 3)/(5 - 2) = (6)/(3) = 2 The limit is 2. e) Step 1: Substitute x=-1 into the function. _x -1 (x^2 - 4)/(x^2 + 3) = ((-1)^2 - 4)/((-1)^2 + 3) Step 2: Simplify the expression. = (1 - 4)/(1 + 3) = (-3)/(4) The limit is -(3)/(4). f) Step 1: Substitute y=0 into the function. _y 0 (3y)/(4y + 3) = (3(0))/(4(0) + 3) Step 2: Simplify the expression. = (0)/(0 + 3) = (0)/(3) = 0 The limit is 0. g) Step 1: Substitute x=1 into the function. _x 1 (5x + 1)/(x - 1) The numerator approaches 5(1) + 1 = 6. The denominator approaches 1 - 1 = 0. Since the numerator is non-zero and the denominator approaches zero, the limit does not exist. Step 2: Check left and right-hand limits to determine behavior. As x 1^+, x-1 > 0, so (5x+1)/(x-1) (6)/(0^+) = +. As x 1^-, x-1 < 0, so (5x+1)/(x-1) (6)/(0^-) = -. Since the left and right-hand limits are not equal, the limit does not exist. The limit is Does not exist. h) Step 1: Substitute t=-1 into the function. _t -1 (10 - 9t^2)/(3t^2 + t - 7) = (10 - 9(-1)^2)/(3(-1)^2 + (-1) - 7) Step 2: Simplify the expression. = (10 - 9(1))/(3(1) - 1 - 7) = (10 - 9)/(3 - 1 - 7) = (1)/(2 - 7) = (1)/(-5) The limit is -(1)/(5). i) Step 1: Substitute x=-2 into the function. _x -2 (x^3 + 8)/(x^2 + 8x + 12) Substituting x=-2 gives ((-2)^3 + 8)/((-2)^2 + 8(-2) + 12) = (-8+8)/(4-16+12) = (0)/(0), which is an indeterminate form. Step 2: Factorize the numerator and denominator. The numerator is a sum of cubes: x^3 + 8 = (x+2)(x^2 - 2x + 4). The denominator is a quadratic: x^2 + 8x + 12 = (x+2)(x+6). Step 3: Rewrite the limit and cancel the common factor (x+2). _x -2 ((x+2)(x^2 - 2x + 4))/((x+2)(x+6)) = _x -2 (x^2 - 2x + 4)/(x+6) Step 4: Substitute x=-2 into the simplified expression.