Final answer:
The limit of x²cos(1/x) as x approaches 0 is zero. Despite the rapidly oscillating cosine function, the x² term dominates near zero, forcing the overall function to approach zero. Option number C is correct.
Step-by-step explanation:
The question asks for the limit as x approaches 0 of the function x²cos(1/x). To find this limit, we need to evaluate how the function behaves as x becomes very small. Since the cosine function oscillates between -1 and 1, and as x approaches zero, 1/x becomes very large, the cosine part will oscillate more rapidly. However, the x² term will approach zero, and any number multiplied by zero is zero. Thus, the limit is controlled by the x² term.
Considering the oscillation of cosine, the maximum and minimum values the function x²cos(1/x) can take are x² and -x², respectively. Therefore, as x approaches zero, both these extremes approach zero, ensuring the function's limit is zero. This means that the limit is not dependent on the oscillation caused by the cosine function because the x² term will shrink to zero regardless of the cosine value.
Thus, the correct answer to the problem is Option C: 0.