Mike May
The intuitive definition of limit is that  for any y-range around a limit point there is an x-range  so the graph exits through the sides of the box rather than though the top or bottom. One can make a similar definition for differentiability with a "bow tie" or double cone drawn around the proposed tangent line at a point. No matter how small an epsilon variation we allow in the slope, we can find a delta so the curve escapes the sides rather than the top or bottom of the double cone. One practical way to get an idea if a function is differentiable at a point is to find values of delta that work when epsilon in .1, .01, and .001. This applet lets you look at a variety of functions.  The value c, where we are checking differentiability, and m, our candidate for the derivative, can be set by a text box on the left hand panel. The values for and , the x and y range of the window are set in the right window. Formally, we have a limit if for every we can find a so the graph goes out the sides rather than the top or bottom of the double cone 
The function choice slider lets you either consider a preloaded function or one of your own construction. These are the preset functions along with features to examine. 1) A polynomial.  It is differentiable everywhere, but the width of a good window depends on the value for c as well as the the amount of variation we allow in the slope. 2) A straight line.  Once again we have differentiable everywhere, but the width of a good window does not depend on the value of c. 3) A trig function, sin(x).   4) sin(1/x).  Not continuous at c=0, so not differentiable. 5) Absolute value, with a corner at x=0. We can only trap one side in a narrow cone. 6) cube root of x.  The tangent line is vertical. Any attempted tangent line fails.. 7) x sin(1/x). Continuous but not differentiable. We keep alternating between different tangent lines. 8) x^2 sin(1/x). Differentiable, even if it keeps wiggling. 9) (x^2)^(1/3). A cusp 10) User choice function: This allow you to enter your favorite function. It should be noted that the size on delta and epsilon is limited so the box is visible on both views. To use a small delta or epsilon, the values of xmin, xmax, ymin, and ymax may need to be adjusted.