Finding if the limit does not exist

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I just want to know, in calculating limits, when I do direct substitution, and it gives 3/0 instead of 0/0, does it mean for sure that the limit does not exist?

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1 Answer

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Yes, it does mean that. Suppose $b_n\to 0$ and $c_n:=\frac{a_n}{b_n}\to L\in \mathbb R$. Then $$\lim_{n\to\infty} a_n = \lim_{n\to\infty}b_n c_n=0\cdot L=0.$$

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