Final answer:
The limit of the function (7x⁵ + 5x² + 12x³) / (x⁵ + 4x) as x approaches 0 is 5, after canceling out common factors and evaluating the simplified expression at x = 0.
Step-by-step explanation:
The question asks for the limit as x approaches 0 of the function (7x⁵ + 5x² + 12x³) / (x⁵ + 4x). To find the limit, we can observe the behavior of the function as x gets close to 0. Since we are dealing with a ratio of polynomials, we can simply plug in x = 0 into the numerator and denominator since they are both of the same degree.
By substituting x = 0, we get:
(7(0)⁵ + 5(0)² + 12(0)³) / ((0)⁵ + 4(0)) = 0 / 0, which is an indeterminate form. However, because each term in the numerator and the denominator contains a factor of x, we can simplify the function:
(7x⁵ + 5x² + 12x³) / (x⁵ + 4x) = (x²(7x³ + 5 + 12x)) / (x(x⁴ + 4)).
After canceling out the common x term, we have:
(7x³ + 5 + 12x) / (x⁴ + 4).
Now, we can safely substitute x = 0 without getting an indeterminate form:
(7(0)³ + 5 + 12(0)) / ((0)⁴ + 4) = 5 / 4 = 5.
The limit of the function as x approaches 0 is 5, so the correct answer is (b) 5.
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