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 evaluations of the limits. i. Evaluate _x 1 (x^8 - 1)/(x^10) - 1 Step 1: Check the form of the limit. If we substitute x=1 into the expression, we get (1^8 - 1)/(1^10) - 1 = (0)/(0), which is an indeterminate form. This means we can apply L'Hôpital's Rule. Step 2: Apply L'Hôpital's Rule. Take the derivative of the numerator and the derivative of the denominator with respect to x. Derivative of the numerator: (d)/(dx)(x^8 - 1) = 8x^7 Derivative of the denominator: (d)/(dx)(x^10 - 1) = 10x^9 Step 3: Evaluate the limit of the new expression. _x 1 (8x^7)/(10x^9) Substitute x=1: (8(1)^7)/(10(1)^9) = (8)/(10) = (4)/(5) The limit exists and is: (4)/(5) ii. Evaluate _x -3 (x^2 - 9)/(x^2 + 2x - 3) Step 1: Check the form of the limit. If we substitute x=-3 into the expression, we get ((-3)^2 - 9)/((-3)^2 + 2(-3) - 3) = (9 - 9)/(9 - 6 - 3) = (0)/(0), which is an indeterminate form. We can apply L'Hôpital's Rule or factor the expression. Let's use factoring for this one. Step 2: Factor the numerator and the denominator. Numerator: x^2 - 9 = (x-3)(x+3) Denominator: x^2 + 2x - 3 = (x+3)(x-1) Step 3: Substitute the factored expressions back into the limit and simplify. _x -3 ((x-3)(x+3))/((x+3)(x-1)) Since x -3, x ≠ -3, so we can cancel the (x+3) term. _x -3 (x-3)/(x-1) Step 4: Evaluate the limit by direct substitution. Substitute x=-3: (-3-3)/(-3-1) = (-6)/(-4) = (3)/(2) The limit exists and is: (3)/(2) Drop the next question!