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