APFIVE Limit at Infinity with an Oscillating Function
A student evaluates the limit $$\lim_{x\to\infty} \frac{x+\sin(x)}{x}$$. Noticing that $$\sin(x)$$ oscillates, she erroneously concludes that the limit does not exist. Identify the mistake in her reasoning.
A
She should have applied L’Hôpital’s rule to resolve the oscillation.
B
The mistake is underestimating the damping effect of dividing by x; since $$\frac{\sin(x)}{x} \to 0$$, the limit is 1.
C
There is no mistake; the oscillation of $$\sin(x)$$ means the limit does not exist.
D
The error is that the limit should be computed as infinity because of the x term.
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