Laboratory 2 
Limits of Functions
Name:

Functions

epsilonDelta[g_,a_,L_]:=Module[{x,f,x0,y0},f[x_]=g[x];x0=a;y0=L;
Manipulate[
GraphicsRow[
{Plot[f[x],{x,x0-delta,x0+delta},PlotRange->{y0-eps,y0+eps},ImagePadding->15,ImageSize->300],
Show[{Plot[f[x],{x,x0-0.6,x0+0.6},PlotRange->{y0-0.6,y0+0.6},ImageSize->300],
Graphics[{{Red,Line[{{x0-0.65,y0-eps},{x0+0.65,y0-eps}, {x0+0.65,y0+eps},{x0-0.65,y0+eps}}]},
{Red,Line[{{x0-delta,y0-0.65},{x0-delta,y0+0.65},{x0+delta,y0+0.65},{x0+delta,y0-0.65}}]}}]}]}],
{{eps,0.5,"ϵ"},0.5,10^(-7)},{{delta,0.5,"δ"},0.5,10^(-7)},LabelStyle->Medium]
]

Experiment 2.1

Determine whether or not the following functions are continuous at x=a using the epsilonDelta command.   Make sure to show a plot that explains what’s going on and state whether the function is continuous or not. 
(a) f(x)=((|x-2|)/(x-2)), a=2,  if  f(2)=0.

(b) g(x)=x sin(1/x), a=0,  if  g(0)=0.

(c) h(x) =((sin x)/x), a=0, if  h(0) = 1.

Experiment 2.2

Use the Limit command to find the following limits.  Check to see what is happening by graphing the functions at the point in question:

1.   (lim_(x->0))((sin(x))/x)

2.  (lim_(x->0))((sin(x)-x)/(x^3))