Is saying 'limit does not exist' the same as saying 'limiting value is $\pm \infty $'? Would it be okay to write $\displaystyle\lim_{x\to 0}\sin {\left (\frac{1}{x}\right )}=\infty$ when the limit clearly does not exist? What I understand is that whenever it is said that 'limit does not exist', it is meant as 'limit does not exist finitely'.
 A: 
Would it be okay to write $\displaystyle\lim_{x\to 0}\sin {\frac{1}{x}}=\infty$ when the limit clearly does not exist?

No, it would not.
Saying that a function tends to infinity, or as a limit:
$$\lim_{x \to a} f(x) = +\infty$$has a very specific meaning, namely that $f(x)$ gets arbitrarily large when $x$ gets close to $a$. You can look up the formal definition of this.
Another example; the limit
$$\lim_{x \to +\infty} \sin x$$
clearly does not exist (as $\sin x$ keeps oscillating between $-1$ and $1$), but it does not tend to infinity.

What I understand is that whenever it is said that 'limit does not exist', it is meant as 'limit does not exist finitely'.

True, but no finite limit is not the same as an infinite limit; see the example above. Tending to $\pm \infty$ is one way of not having a finite limit, but it is not the only one.
A: No, these are not the same thing. Note that $|\sin{(x)}|\leq 1$ for all $x\in \mathbb{R}$. To say that $$\lim_{x\rightarrow \infty}\sin x =\infty$$ means given any $M>0$ we can find $c\in \mathbb{R}$ such that for all $x>c$ we have $\sin(x)>M$. That is, if we look far enough along the reals we can make our function as large as we like.
However, to say the limit does not exist is to say that for any number $L$, there is an $\epsilon >0$ such that for all $\delta>0$ you can find $x\in \mathbb{R}$ such that $|x-a|<\delta$ but $|\sin{(x)} - L|>\epsilon$.
A: No, this does not mean that the limit does not exist. This means that the limit is greater than any real number, eg $\lim\limits_{x \rightarrow \infty} x = \infty$ as the limit exists, although it is not finite. For your case, you have to say that the limit does not exist as this is not equivalent to the limit being $\infty$.
A: Definetly not, limit does not exist COULD MEAN limit isn't the same when approached from different point of views. these point of view could be different lines in when calculating limit in R^3 or different -x and +x directions when calculating limit in R^2.
$$
\lim_{x{\to+3}}f(x) = 3\\
\lim_{x{\to+3}}f(x) = 5 \\ 
$$
Above could be the case, when the function is distracted, since both cases aren't equal each other the limit does not exist. See, limit doesn't have to be +infty or -infty
A: let $f(x)=\sin\left(\frac{1}{x}\right)$ and
$$a_n=\frac{1}{2n\pi+\frac{\pi}{2}}$$
and
$$b_n=\frac{1}{2n\pi}$$
we have
\begin{align}
  & \underset{n\to \infty }{\mathop{\lim }}\,f({{a}_{n}})=1 \\ 
 & \underset{n\to \infty }{\mathop{\lim }}\,f({{b}_{n}})=0 \\ 
\end{align}
Thus we can say this limit does not exist. 
