Limit at Infinity with an Oscillating Function

Question

When evaluating the limit $$\lim_{x	o\infty} \frac{x+\sin(x)}{x}$$, a student concludes that the limit does not exist because the function $$\sin(x)$$ oscillates. Which statement best explains the error in the student's reasoning?

Accepted Answer

The mistake is underestimating the damping effect of dividing by x; since $$\frac{\sin(x)}{x} 	o 0$$, the limit is 1.

Answer

There is no mistake; the oscillation of $$\sin(x)$$ means the limit does not exist.

Answer

She should have applied L'Hôpital's rule to resolve the oscillation.

Answer

The error is that the limit should be computed as infinity because of the x term.

Limit at Infinity with an Oscillating Function

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