Question

Which method is used to evaluate the limit limx→[infinity] 4x⁵cos(13x⁵π/2)?

a) L'Hopital's Rule
b) Product Rule
c) Chain Rule
d) Quotient Rule

Answer 1

To evaluate the limit lim(x→∞) 4x⁵cos(13x⁵π/2), we can use L'Hopital's Rule because it is an indeterminate form. After applying L'Hopital's Rule and simplifying, the limit evaluates to infinity. The correct answer is a) L'Hopital's Rule.

Step-by-step explanation:

To evaluate the limit lim(x→∞) 4x⁵cos(13x⁵π/2), we can use L'Hopital's Rule because it is an indeterminate form. L'Hopital's Rule states that if we have a limit of the form 0/0 or ∞/∞, we can differentiate both the numerator and denominator until we get a determinate form. In this case, we can take the derivatives of 4x⁵ and cos(13x⁵π/2) to simplify the expression.

Applying L'Hopital's Rule, we calculate the derivatives and get lim(x→∞) 20x⁴(-65π/2)sin(13x⁵π/2). Then, we can simplify further by noting that as x approaches infinity, sin(13x⁵π/2) will oscillate between -1 and 1. Therefore, the limit evaluates to lim(x→∞) -65π/2 * 20x⁴.

Since the term -65π/2 is a constant, it can be factored out of the limit and we are left with -65π/2 * lim(x→∞) 20x⁴. As x approaches infinity, the term 20x⁴ grows without bound, so the limit is infinity. Therefore, the answer is infinity.