ArticleEpsilon-Delta Tool

Suppose you suspect that \[\lim_{x\rightarrow c}\,f(x)=L\]

This tool allows you to estimate values of \(\delta\) which guarantee that the function \(f(x)\) is within \(\epsilon\) units of the limit L for all values of \(x\) within \(\delta\) of \(c\) (i.e. \(|x-c|\lt\delta\)).

Modify the definition of f to define the function.
Modify the value of \(c\) to change the \(x\)-value where the limit is being estimated.
Modify the value of \(L\) to change the proposed limit.
Modify the value of epsilon to reflect the vertical tolerance.
Modify the value of delta to modify the horizontal tolerance.

If the blue curve passes through the pink boxes, your value for delta needs to be smaller.

If no value of delta keeps the the blue curve out of the pink boxes, then the value of your limit (\(L\)) is wrong.

{{{ @interact def delta_epsilon(f = input_box(default=(x^2-x)/(x-1)), c=input_box(default=1), L = input_box(default=1), epsilon=input_box(label="epsilon",default=0.1), delta=input_box(label="delta",default=0.1)): xm = c-2*delta xM = c+2*delta f_left_plot = plot(f,xm,c-delta/20,thickness=2) f_right_plot = plot(f,c+delta/20,xM,thickness=2) epsilon_line_1 = line([(xm,L-epsilon),(xM,L-epsilon)], rgbcolor=(0.5,0.5,0.5),linestyle='--') epsilon_line_2 = line([(xm,L+epsilon),(xM,L+epsilon)], rgbcolor=(0.5,0.5,0.5),linestyle='--') ym = L-2*epsilon yM = L+2*epsilon bad_region_1 = polygon([(c-delta,L+epsilon),(c-delta,yM),(c+delta,yM),(c+delta,L+epsilon)], rgbcolor=(1,0.6,0.6)) bad_region_2 = polygon([(c-delta,L-epsilon),(c-delta,ym),(c+delta,ym),(c+delta,L-epsilon)], rgbcolor=(1,0.6,0.6)) cL_point = point((c,L),rgbcolor=(1,0,0),pointsize=20) delta_line_1 = line([(c-delta,ym),(c-delta,yM)],rgbcolor=(0.5,0.5,0.5),linestyle='--') delta_line_2 = line([(c+delta,ym),(c+delta,yM)],rgbcolor=(0.5,0.5,0.5),linestyle='--') (f_left_plot +f_right_plot +epsilon_line_1 +epsilon_line_2 +delta_line_1 +delta_line_2 +cL_point +bad_region_1 +bad_region_2).show(xmin=xm,xmax=xM,ymin=ym, ymax=yM, axes=False,frame=True, gridlines=[[c-1.5*delta, c-0.5*delta, c+0.5*delta, c+1.5*delta], [L-1.5*epsilon,L-0.5*epsilon,L+0.5*epsilon,L+1.5*epsilon]]) /// }}}