Final answer:
The given limit equation is correct because, after factoring the numerator and canceling the common factor with the denominator, we are left with the function x + 6, which approaches the same limit as the simplified function does as x approaches 5.
Step-by-step explanation:
The question at hand relates to understanding the limit of a function as x approaches a particular value. In the provided equation, lim x → 5 x2 + x - 30 / (x - 5), the function has a factor that can be canceled out to simplify it to lim x → 5 (x + 6). To determine why the equation is correct, factor the quadratic equation in the numerator.
Factoring x2 + x - 30 gives (x + 6)(x - 5). Once this is done, the (x - 5) in the denominator can be canceled out with the (x - 5) factor in the numerator, leaving us with x + 6. This simplification is valid for all x except for x = 5, where the original denominator becomes zero, which is not permissible in mathematics. However, since limits allow us to explore the behavior of a function as it approaches a point without necessarily reaching that point, choice (b) is the most accurate answer to why the equation is correct as it addresses the continuity and behavior at points other than x = 5. Since the equation holds for all x ≠ 5, both sides of the equation approach the same limit as x → 5.