For the function $f$, given by$$f(x) = \begin{cases} a & (x=0) \\ \frac {\sinh(x)}{\sinh(2x)} & (x\neq0)\end{cases}$$
Provided the limits exist, I need to determine, $$\lim_{x \to 0}f(x)$$
I know that $f(x)$ is undefined at $x=0$. Furthermore, I've determined that $\lim_{x \to 0} \frac {\sinh(x)}{\sinh(2x)} = \frac{1}{2}$.
Now, as far as my confusion is concerned. If the question only asked for $\lim_{x \to 0} \frac {\sinh(x)}{\sinh(2x)}$, I would simply show how I computed $\frac{1}{2}$.
However, with the additional case where $f(x) = a$, alongside the one-sided curly bracket notation, I'm not sure how to approach this problem. I cannot find any literature to enlighten me on how to use this notation in the context of limits, so I apologise if this is a duplicate. Perhaps this is because I don't know what name to give this one-sided curly bracket notation.
I do however have one idea on how to approach this:
Which is to show that $$\lim_{x \to 0} \frac {\sinh(x)}{\sinh(2x)} = \frac{1}{2} =a$$
Am I correct in my interpretation on how to handle this notation in the context of computing limits? Or am I missing something? Either way, how can this notation be explained in precise terms in this context? At the moment, it is fairly vague in my mind.
Limit of $f(x)$ when $x \to 0$ is about a neighborhood of $0$ which excludes zero itself.
You also mentioned
No, according to the definition $$f(0)=a$$
Limit is not about $f(0)$ at all. Even if $f(x=0)$ is undefined, it does not mean that this limit does not exists.
When saying
$$\lim _{x\to c}f(x)=L\iff (\forall \varepsilon >0,\,\exists \ \delta >0,\,\forall x\in D,\,0<|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon )$$
This definition clearly means $$x\ne c$$
The only case when it is important is when they state that $f$ is continuous at $0$. Then it means its limit and its value are equal leading to $a=\frac12$.