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Find the limit as n approaches infinity of cos(x)/x³?

A) 0
B) 1
C) [infinity]
D) The limit does not exist.

1 Answer

2 votes

Final answer:

The limit as n approaches infinity of cos(x)/x³ is 0 because while the numerator, cos(x), is bounded, the denominator, x³, grows without bound. Hence, the fraction approaches 0. the correct option is: A) 0.

Step-by-step explanation:

The question asks to find the limit as n approaches infinity of the function cos(x)/x³. In mathematics, we analyze limits to understand the behavior of functions as the variables approach certain values. As n approaches infinity, the denominator x³ becomes infinitely large. Since the absolute value of the cosine function is bounded between -1 and 1, the effect of the increasing x³ in the denominator will dominate, driving the value of the entire fraction towards zero.

Therefore, regardless of the oscillatory nature of the cosine function, the limit as n approaches infinity of cos(x)/x³ is 0. This is because the growth rate of the cubic term x³ is much faster than the oscillation of cos(x), thus rendering the oscillations insignificant in the long run.

The limit of this function is an example of a problem in calculus, a branch of mathematics that deals with continuous change and includes the study of limits, derivatives, integrals, and infinite series.

For the final answer, the correct option is: A) 0.

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