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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The limit evaluates to 1/2. Steps: We need to evaluate the limit: L = _x 0 ( x - x)/(x^3) First, substitute x=0 into the expression: Numerator: (0) - (0) = 0 - 0 = 0 Denominator: 0^3 = 0 Since this is an indeterminate form (0)/(0), we can use L'Hôpital's Rule or Taylor series expansions. Method 1: Using L'Hôpital's Rule Apply L'Hôpital's Rule by taking the derivative of the numerator and the denominator. 1st Application: Derivative of numerator: (d)/(dx)( x - x) = ^2 x - x Derivative of denominator: (d)/(dx)(x^3) = 3x^2 L = _x 0 (^2 x - x)/(3x^2) Substitute x=0: Numerator: ^2(0) - (0) = 1^2 - 1 = 0 Denominator: 3(0)^2 = 0 Still an indeterminate form (0)/(0). 2nd Application: Derivative of numerator: (d)/(dx)(^2 x - x) = 2 x( x x) - (- x) = 2^2 x x + x Derivative of denominator: (d)/(dx)(3x^2) = 6x L = _x 0 (2^2 x x + x)/(6x) Substitute x=0: Numerator: 2^2(0) (0) + (0) = 2(1)^2(0) + 0 = 0 Denominator: 6(0) = 0 Still an indeterminate form (0)/(0). 3rd Application: Derivative of numerator: (d)/(dx)(2^2 x x + x) Using the product rule for 2^2 x x: 2[(2 x( x x)) x + ^2 x(^2 x)] + x = 2[2^2 x ^2 x + ^4 x] + x = 4^2 x ^2 x + 2^4 x + x Derivative of denominator: (d)/(dx)(6x) = 6 L = _x 0 (4^2 x ^2 x + 2^4 x + x)/(6) Now, substitute x=0: Numerator: 4^2(0) ^2(0) + 2^4(0) + (0) = 4(1)^2(0)^2 + 2(1)^4 + 1 = 0 + 2 + 1 = 3 Denominator: 6 L = (3)/(6) = (1)/(2) Method 2: Using Taylor Series Expansions Recall the Maclaurin series expansions for x and x around x=0: x = x - (x^3)/(3!) + (x^5)/(5!) - = x - (x^3)/(6) + O(x^5) x = x + (x^3)/(3) + (2x^5)/(15) + = x + (x^3)/(3) + O(x^5) Substitute these into the limit expression: L = _x 0 ((x + x^3)/(3) + O(x^5)) - (x - (x^3)/(6) + O(x^5))x^3 L = _x 0 (x + x^3)/(3) - x + (x^3)/(6) + O(x^5)x^3 L = _x 0 ((1)/(3) + (1)/(6))x^3 + O(x^5)x^3 L = _x 0 ((2)/(6) + (1)/(6))x^3 + O(x^5)x^3 L = _x 0 (3)/(6)x^3 + O(x^5)x^3 L = _x 0 (1)/(2)x^3 + O(x^5)x^3 Divide by x^3: L = _x 0 ((1)/(2) + O(x^2)) As x 0, O(x^2) 0. L = (1)/(2) The final answer is 1/2.