What does differentiable mean for a function?

Hint: In the given question about function differentiability, we need to basically go thoroughly about the concept of differentiability and also about the limits we find in order to find the nature of differentiability.

Complete step-by-step answer:
According to the question, in order to define differentiability of the function let us suppose f be any function such that it is differentiable at a if it has a non-vertical tangent at the corresponding point on the graph where the corresponding point is (a, f(a)).
So, from this we can conclude that the limit would exist which is $\displaystyle \lim_{x \to a}\dfrac{f\left( x \right)-f\left( a \right)}{x-a}$ .
Now, we know that limit exists means that it is the finite value. Also, this limit is said to be the tangent line slope.
So, we can say that whenever this limit exists as a finite value, we can say that the function is differentiable at a or we can say that it is the derivative of f at a and is denoted by$f'\left( a \right)$ or $\dfrac{df\left( a \right)}{dx}$.
Also, we say that if at some point the above defined limit does not exist, we say that the given function is not differentiable at that point. Here, limit does not exist means that limit is infinite which is the case when the tangent is vertical. This case may occur when the function is discontinuous or when there are two one sided limits.
So, in order to check the differentiability of the function we need to check the limits which is given by $\displaystyle \lim_{x \to a}\dfrac{f\left( x \right)-f\left( a \right)}{x-a}$at any point a.

Note: In this question we need to check about the differentiability of the function so for that we need to check the limits and for different situations of limits we have different choices and we need to do verify all these cases and identify whether or not the kind of limit asked is satisfied by our limit or not.