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 practical way to get an idea if a point is a limit is to find values of delta [br]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 taking the limit, and L,[br]our candidate for the limit, can be set either by a slider or 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 box 

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 parabola.  We have a limit at each point but the width of a good window depends on the value for c as well as the height of the box.[br]2) A straight line.  Once again we have a limit at each point, but the width of a good window does not depend on the value of c.[br]3) A parabola with a hole at x=1.  Removable holes do not have an impact on limits.[br]4) sin(x)/x.  This is a standard limit that will need to get evaluated.  It is worthwhile noticing how nice the graph is.[br]5) sin(1/x)+1.  If we focus on c=0, the graph is going up and down so fast that we cannot find a limit.  At any other point we can find a limit.[br]6) x sin(1/x) + 1.  This is a modification of the previous example.  It still wiggles, but the wiggles get smaller so we can have a limit.[br]7) This function uses a "greatest integer function" in its construction.  It has a lot of breaks.[br]8) This function goes off to positive infinity when c=0 so depending on our convention either we cannot have a limit, or the limit is positive infinity.[br]9) This modifies the previous example to go to positive and negative infinity at the same time.[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.