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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SECTION B (20 Marks) I. Consider the function whose formula is f(x) = (16-x^4)/(x^2-4). Find f(2) and _x -2 f(x). Step 1: Find f(2). Substitute x=2 into the function: f(2) = (16-(2)^4)/((2)^2-4) = (16-16)/(4-4) = (0)/(0) Since this is an indeterminate form, the function is undefined at x=2. f(2) is undefined Step 2: Find _x -2 f(x). First, attempt to substitute x=-2 into the function: f(-2) = (16-(-2)^4)/((-2)^2-4) = (16-16)/(4-4) = (0)/(0) This is an indeterminate form, so we need to simplify the function by factoring. Step 3: Factorize the numerator and denominator. The numerator is a difference of squares: 16-x^4 = (4-x^2)(4+x^2). The denominator is also a difference of squares: x^2-4. So, we can write f(x) as: f(x) = ((4-x^2)(4+x^2))/(x^2-4) Notice that (4-x^2) = -(x^2-4). Substitute this into the expression: f(x) = (-(x^2-4)(4+x^2))/(x^2-4) Step 4: Simplify the function. For x^2-4 ≠ 0 (i.e., x ≠ 2 and x ≠ -2), we can cancel the common factor (x^2-4): f(x) = -(4+x^2) Step 5: Evaluate the limit. Now, find the limit as x -2 for the simplified function: _x -2 f(x) = _x -2 -(4+x^2) Substitute x=-2 into the simplified expression: = -(4+(-2)^2) = -(4+4) = -8 The final answers are: f(2) is undefined _x -2 f(x) = -8 That's 2 down. 3 left today — send the next one.