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Question:

True/False?

$$\lim_{x\to 0} \frac{sin(x^2)}{x^2}sin(\frac{1}{x}) = 1$$

Discussion:

As $$x$$ goes to zero, $$x^2$$ also goes to zero, and consequently, since $$\lim_{y\to 0} \frac{sin(y)}{y}=1$$ we have $$\lim_{x\to 0} \frac{sin(x^2)}{x^2}= 1$$. If the statement were true, then $$\lim_{x\to 0}sin(\frac{1}{x}) = 1$$, which is false. In fact, $$\lim_{x\to 0}sin(\frac{1}{x})$$ does not exist.

So the statement is False.