Visual difference between limits and derivatives? I've heard of limit, continuity and derivative explained algebraically but I was wondering about visually too. I was wondering if someone could explain the purpose/difference of limit, continuity and derivative?
I'm sorry I forgot the image!
 A: Take for example $f(x)=x+1$. Then $Lim_{x \rightarrow 1}f(x)=Lim_{x \rightarrow 1}(x+1)=1+1=2$. Note also that $f(1)=2$ (i.e., by substituting $1$ for $x$ in the function $f(x)$, we get $2$).
It is possible for the limit of a function to exist, without the value of the function existing. For instance, take $m(x)=\frac{x^2-4}{x-2}$. Now, $Lim_{x \rightarrow 2}m(x)=Lim_{x \rightarrow 2}\frac{(x-2)(x+2)}{x-2}=Lim_{x \rightarrow 2}(x+2)=2+2=4$, but $m(2)$ does not exist.
Definition: A function $f(x)$ is said to be continuous at a point $x=x_0$ if the following three conditions are satisfied: (i) the value of the function exists ($f(x_0)$ exists); (ii) the limit of the function exists ($Lim_{x \rightarrow x_0} f(x)$ exists); (iii) the value of the function=the limit of the function ($Lim_{x \rightarrow x_0}f(x)=f(x_0)$). 
From the example above, the function $f(x)$ is continuous at $x=1$ (because it satisfies (i)-(iii) above). Note that $g(x)=\frac{1}{x-1}$ is not continuous at $x=1$ because the limiting value at $x \rightarrow 1$ does not exist. Can you also see that $g(1)$ does not exist? On the other hand, though $Lim_{x \rightarrow 2}m(x)$ exists, $m(x)$ is not still continuous at $x=2$ because $m(2)$ does not exist.
Remark: A very easy way to visualise a continuous function is a function you can draw its graph without lifting up your hand from the paper. So, if you happen to lift your hand anywhere while drawing the graph, then those points are the point of discontinuities of the function. 
Continuity is a big thing that is now reduced to a naive/smaller thing through the remark above. In particular, whenever a function is continuous at a point, by the definition of continuity above, we are sure that the limit also exists at that point.
Derivative existing is a bigger condition than continuity. On the other hand, a function is continuous at a point $x_0$ implies that the limit of the function exists at that same point $x_0$. 
In summary, $h(x)$ is differentiable at $x=x_0$ $\Rightarrow$ $h(x)$ is continuous at $x_0$ $\Rightarrow$ $Lim_{x \rightarrow x_0}h(x)$ exists. But the converse of any of the statements is not necessarily true.
