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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You're on a roll — here are the solutions to the limit problems. 3. We need to find the value of k given the equality: _x 1 (x^4 - 1)/(x - 1) = _x k (x^3 - k^3)/(x^2 - k^2) Step 1: Evaluate the left-hand side (LHS) limit. The LHS is in the form _x c (x^n - c^n)/(x - c) = n c^n-1. Here, c=1 and n=4. _x 1 (x^4 - 1^4)/(x - 1) = 4 · (1)^4-1 = 4 · 1^3 = 4 Step 2: Evaluate the right-hand side (RHS) limit. The RHS is an indeterminate form (0)/(0) when x=k. We can rewrite it by dividing the numerator and denominator by (x-k): _x k (x^3 - k^3)/(x^2 - k^2) = _x k (x^3 - k^3)/(x - k)(x^2 - k^2)/(x - k) Apply the limit formula _x c (x^n - c^n)/(x - c) = n c^n-1 to both the numerator and the denominator. Numerator: _x k (x^3 - k^3)/(x - k) = 3k^3-1 = 3k^2. Denominator: _x k (x^2 - k^2)/(x - k) = 2k^2-1 = 2k. So, the RHS becomes: (3k^2)/(2k) Assuming k 0, we can simplify this to (3k)/(2). If k=0, the RHS would be _x 0 (x^3)/(x^2) = _x 0 x = 0, which is not equal to the LHS (4). Thus, k 0. Step 3: Equate the LHS and RHS and solve for k. 4 = (3k)/(2) 8 = 3k k = (8)/(3) The value of k is (8)/(3). 4. We need to find the possible values of a given the equality: _x -a (x^9 + a^9)/(x + a) = 9 Step 1: Rewrite the limit expression to match the standard form. The limit is an indeterminate form (0)/(0) when x=-a. We use the formula _x c (x^n - c^n)/(x - c) = n c^n-1. Since n=9 is an odd number, a^9 = -(-a)^9. So, x^9 + a^9 = x^9 - (-a)^9. Also, x+a = x - (-a). Therefore, the limit can be written as: _x -a (x^9 - (-a)^9)/(x - (-a)) Step 2: Apply the limit formula. Here, c = -a and n = 9. The limit is 9 · (-a)^9-1 = 9 · (-a)^8. Since any non-zero number raised to an even power is positive, (-a)^8 = a^8. So, the limit evaluates to 9a^8. Step 3: Equate the result to the given value and solve for a. 9a^8 = 9 a^8 = 1 To find a, we take the eighth root of both sides. Since the exponent is even, there will be two real solutions (positive and negative). a = ± [8]1 a = ± 1 The possible values of a are ± 1. Got more? Send 'em!