Using sequences to show that limits of certain functions as certain values are approached do not exist I vaguely recall from undergraduate analysis that there is a relationship that allows
sequences to be used to show that certain limits don't exist. For example, I recently
came across a limit problem, specifically the limit of $\sin(1/x)$ as $x$ approaches zero.
The person who posted the solution used sequences to show that this limit does not exist.
I was wondering if someone would be kind enough to refresh my memory and generalize this
process along with any theorems that may be necessary or perhaps refer me to another internet resource that would do so, as I was
foolish enough to sell my analysis text, and I don't believe a basic undergraduate calculus text would cover this topic.

Thanks
For any fixed $a\in[-1,1]$, the equation $f(x)=\sin(1/x)=a$ has infinitely many solutions on every interval $(0,\delta)$, from which you can construct a sequence $x_n\to 0$ such that $\lim_{n\to\infty} f(x_n)=a$. Since you can reach any $a\in[-1,1]$, the limit as a function does not exist. This applies to any oscillating singularity.
For any fixed $a\in[-1,1]$, the equation $f(x)=\sin(1/x)=a$ has infinitely many solutions on every interval $(0,\delta)$, from which you can construct a sequence $x_n\to 0$ such that $\lim_{n\to\infty} f(x_n)=a$. Since you can reach any $a\in[-1,1]$, the limit as a function does not exist. This applies to any oscillating singularity.