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 image contains two distinct questions. I will address them separately. Question 1: Evaluate the limit Step 1: Identify the limit expression. The limit to evaluate is: _x 2 (x^2 + 2x - 12)/(x^2 - 2x) Step 2: Attempt direct substitution. Substitute x=2 into the expression: Numerator: 2^2 + 2(2) - 12 = 4 + 4 - 12 = -4 Denominator: 2^2 - 2(2) = 4 - 4 = 0 Since direct substitution results in (-4)/(0), the limit does not exist in the form of a finite number. It approaches either or -. Step 3: Analyze the behavior of the function as x approaches 2. Let f(x) = (x^2 + 2x - 12)/(x^2 - 2x). As x 2, the numerator approaches -4. The denominator x^2 - 2x = x(x-2). As x 2^+, x-2 is a small positive number, so x(x-2) is a small positive number. Thus, f(x) (-4)/(small positive) -. As x 2^-, x-2 is a small negative number, so x(x-2) is a small negative number. Thus, f(x) (-4)/(small negative) +. Since the left-hand limit and the right-hand limit are not equal, the limit does not exist. The limit is: Does not exist Question 2: Is this function continuous at x=27? Step 1: Identify the function and the point of interest. The function is f(x) = (x^2 + 2x - 12)/(x^2 - 2x). We need to check its continuity at x=27. Step 2: Check the conditions for continuity. A function f(x) is continuous at a point x=a if three conditions are met: 1. f(a) is defined. 2. _x a f(x) exists. 3. _x a f(x) = f(a). Step 3: Evaluate the function at x=27. Substitute x=27 into the function: f(27) = (27^2 + 2(27) - 12)/(27^2 - 2(27)) f(27) = (729 + 54 - 12)/(729 - 54) f(27) = (783 - 12)/(675) f(27) = (771)/(675) Since the denominator 675 ≠ 0, f(27) is defined. Step 4: Evaluate the limit as x 27. For a rational function, if the denominator is not zero at the point, the limit can be found by direct substitution. _x 27 (x^2 + 2x - 12)/(x^2 - 2x) = (27^2 + 2(27) - 12)/(27^2 - 2(27)) = (771)/(675) The limit exists and is equal to f(27). Step 5: Conclude on continuity. Since f(27) is defined and _x 27 f(x) = f(27), the function is continuous at x=27. The function is continuous at x=27. Yes, the function is continuous at x=27 What's next? Send 'em!