![Find the limiting value of lim x->2 [2x-2]/[x-1]](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1780761507211-acc108854a8bd0f0.png&w=3840&q=75)
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 given limit problems: 1. Find the limiting value of _x 2 \ (2x-2)/(x-1) \ Step 1: Substitute x=2 into the expression. _x 2 \ (2x-2)/(x-1) \ = (2(2)-2)/(2-1) Step 2: Simplify the expression. = (4-2)/(1) = (2)/(1) = 2 The limiting value is 2. 2. Evaluate _x 3 (x-3)/(x^2-5x+6) Step 1: Substitute x=3 into the expression. This results in an indeterminate form (0)/(0). (3-3)/(3^2-5(3)+6) = (0)/(9-15+6) = (0)/(0) Step 2: Factor the denominator. x^2-5x+6 = (x-2)(x-3) Step 3: Rewrite the limit expression and cancel out the common factor (x-3). _x 3 (x-3)/((x-2)(x-3)) = _x 3 (1)/(x-2) Step 4: Substitute x=3 into the simplified expression. = (1)/(3-2) = (1)/(1) = 1 The limit is 1. 3. Evaluate _x 0 (2x^3-3x^2+4x)/(5x^3+2x^2-3x) Step 1: Substitute x=0 into the expression. This results in an indeterminate form (0)/(0). (2(0)^3-3(0)^2+4(0))/(5(0)^3+2(0)^2-3(0)) = (0)/(0) Step 2: Factor out x from both the numerator and the denominator. _x 0 (x(2x^2-3x+4))/(x(5x^2+2x-3)) Step 3: Cancel out the common factor x. = _x 0 (2x^2-3x+4)/(5x^2+2x-3) Step 4: Substitute x=0 into the simplified expression. = (2(0)^2-3(0)+4)/(5(0)^2+2(0)-3) = (0-0+4)/(0+0-3) = (4)/(-3) The limit is -(4)/(3). 4. Evaluate _x 0 ((3x-4)(2x+3)-(5x-2)(x+6))/(x(x^2-9)) Step 1: Substitute x=0 into the expression. This results in an indeterminate form (0)/(0). ((3(0)-4)(2(0)+3)-(5(0)-2)(0+6))/(0(0^2-9)) = ((-4)(3)-(-2)(6))/(0) = (-12 - (-12))/(0) = (0)/(0) Step 2: Expand and simplify the numerator. (3x-4)(2x+3) = 6x^2+9x-8x-12 = 6x^2+x-12 (5x-2)(x+6) = 5x^2+30x-2x-12 = 5x^2+28x-12 Numerator: (6x^2+x-12) - (5x^2+28x-12) = 6x^2+x-12-5x^2-28x+12 = (6x^2-5x^2) + (x-28x) + (-12+12) = x^2-27x Step 3: Factor out x from the simplified numerator. x^2-27x = x(x-27) Step 4: Rewrite the limit expression with the simplified numerator and cancel out the common factor x. _x 0 (x(x-27))/(x(x^2-9)) = _x 0 (x-27)/(x^2-9) Step 5: Substitute x=0 into the simplified expression. = (0-27)/(0^2-9) = (-27)/(-9) = 3 The limit is 3. Send me the next one 📸