Final answer:
f(x) = (x^5 - 32)/(x - 2) is defined for all x except x = 2, where the function has a removable discontinuity. By simplifying the function, the limit as x approaches 2 is found to be 80.
Step-by-step explanation:
The correct answer is option Mathematics and involves the discussion of the definition and limits of a function, specifically f(x) = (x^5 - 32)/(x - 2). This function is defined for all real numbers except at x = 2 because that would result in a division by zero, which is undefined.
To determine the limit as x approaches 2, where the function is not defined, we can simplify the numerator by factoring it as a difference of two powers and then cancel out the common factor with the denominator. The simplified version of this function would be x^4 + 2x^3 + 4x^2 + 8x + 16, which is defined at x = 2. The limit as x approaches 2 is then the value of this simplified function at x = 2, which is 16 + 16 + 16 + 16 + 16 = 80.
The function f(x) = (x^5 - 32)/(x - 2) is defined for all values of x except x = 2. This is because when x = 2, the denominator becomes 0 and division by 0 is undefined. However, for all other values of x, the function is defined and can be evaluated.
To check the limit as x approaches 2, we can use algebraic manipulation. By factoring the numerator, we can rewrite the function as f(x) = [(x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)]/(x - 2). Simplifying, we get f(x) = x^4 + 2x^3 + 4x^2 + 8x + 16. This new expression is defined for x = 2, and we can compute the limit as x approaches 2 by substituting 2 into the expression, which gives us 2^4 + 2(2^3) + 4(2^2) + 8(2) + 16 = 62.
In summary, the function f(x) = (x^5 - 32)/(x - 2) is defined for all values of x except x = 2. The limit of the function as x approaches 2 is 62.