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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stephennaomi63, let's knock this out. The image contains multiple questions. I will solve question 4.a. Question 4.a asks to determine the continuity of the given function f(x) at x=3. The function is defined as: f(x) = (x^3 - 27)/(x^2 - 9) & for x ≠ 3 \\ 5 & for x = 3 For a function to be continuous at a point x=c, three conditions must be met: 1. f(c) must be defined. 2. _x c f(x) must exist. 3. _x c f(x) = f(c). Step 1: Check if f(3) is defined. From the definition of f(x), when x=3, f(x)=5. So, f(3) = 5. The function is defined at x=3. Step 2: Evaluate the limit _x 3 f(x). For x ≠ 3, f(x) = (x^3 - 27)/(x^2 - 9). We can factor the numerator and denominator. The numerator is a difference of cubes: a^3 - b^3 = (a-b)(a^2 + ab + b^2). So, x^3 - 27 = x^3 - 3^3 = (x-3)(x^2 + 3x + 9). The denominator is a difference of squares: a^2 - b^2 = (a-b)(a+b). So, x^2 - 9 = x^2 - 3^2 = (x-3)(x+3). Now, substitute these factored forms into the limit expression: _x 3 (x^3 - 27)/(x^2 - 9) = _x 3 ((x-3)(x^2 + 3x + 9))/((x-3)(x+3)) Since x 3, x ≠ 3, so x-3 ≠ 0. We can cancel the (x-3) terms. _x 3 (x^2 + 3x + 9)/(x+3) Now, substitute x=3 into the simplified expression: (3^2 + 3(3) + 9)/(3+3) = (9 + 9 + 9)/(6) = (27)/(6) Simplify the fraction: (27)/(6) = (9)/(2) So, _x 3 f(x) = (9)/(2). Step 3: Compare _x 3 f(x) with f(3). We found f(3) = 5 and _x 3 f(x) = (9)/(2). Since 5 ≠ (9)/(2), the third condition for continuity is not met. Therefore, the function f(x) is not continuous at x=3. The function f(x) is not continuous at x=3 because _x 3 f(x) ≠ f(3). What's next? Send 'em.