TITLE: When to solve limits using sequences
QUESTION [0 upvotes]: Like the title says when do we solve a limit using sequences? For example consider $\lim_{x\to 0} x \sin\left(\frac {1}{x}\right) $ here we can use either sequences or boundedness of $\sin $ function. But now consider
$$f(x)= \begin{cases} 0 &\text{if $x$ is rational} \\ 1 & \text{if $x$ is irrational.} \end  {cases}
$$. Then find $f (x)_{\lim {x \to 0}} $
Here we can  use only sequences.  How  one can  be sure that solving a particular limit by sequences would be the most efficient way?

REPLY [1 votes]: In general, "using sequences" is never "the most efficient way" because in order to show that $\lim_{x\to a} f(x)= L$ "using sequences" we would have to show that $\lim_{n\to \infty} f(x_n)= L$ for every sequence $\{x_n\}$ such that $x_n\to a$.
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