The following formula defines the function ff. For  Mathematica to learn this function you must place the cursor at the end of the formula and then press  Shift+Enter.  That hold for all cells with commands. For more instructions see my Mathematica page.

ff[x_] := (x/(x + 1))

By changing the formula you can explore another function. See the Examples below.

N[ff[1000]]

The following formula defines ϵ

ϵ = 0.005

Plot[{ff[x], 1}, {x, 0, 1000}, PlotStyle -> {Blue, Black}, 
      PlotRange -> {{100, 200}, { .9, 1.1}}, 
      Epilog -> {{Green, 
            Line[{{0, 1 - ϵ}, {1000, 1 - ϵ}}], 
            Line[{{0, 1 + ϵ}, {1000, 1 + ϵ}}]}}, 
      Axes -> False, Frame -> True]

Now I will automate this process using X(ϵ) from the definiton of limit.

Example 1

The function that we are studing is

ff[x_] := (x/(x + 1))

We calculated in class

Clear[ϵ, X, X0] ;
L = 1;

    
X0 = 0;

    
X[ϵ_] := (1/ϵ)

Now plot what we found. Remember that the blue graph must be between the green horizontal lines for all x > X(ϵ).  I put a vertical red line to mark X(ϵ),

But to get a picture we need to set a specific ϵ.

ϵ = 
        0.000000000000000000000001;

Plot[{1, ff[x]}, {x, X0, 10^25}, PlotStyle -> {Black, Blue}, 
      PlotRange -> {{X[ϵ] - 2, X[ϵ] + 3000}, {L -  .1, 
            L +  .1}}, 
      Epilog -> {{Green, 
            Line[{{X0, L - ϵ}, {1000, L - ϵ}}], 
            Line[{{X0, L + ϵ}, {1000, L + ϵ}}]}, {Red, 
            Line[{{X[ϵ], 0}, {X[ϵ], 100}}]}}, 
      Axes -> False, Frame -> True]

Example 2

The function that we are studing is from the notes.

ff[x_] := (1/(Sqrt[x - 1]))

We calculated in the notes

Clear[ϵ, X, 
        X0] ;

L = 0;

X0 = 2;

X[ϵ_] := (1/(ϵ^2)) + 2

Now plot what we found. Remember that the blue graph must be between the green horizontal lines for all x > X(ϵ).  I put a vertical red line to mark X(ϵ),

But to get a picture we need to set a specific ϵ.

ϵ = 0.1;

Plot[{ff[x], L}, {x, X0, 1000}, PlotStyle -> {Blue, Black}, 
      PlotRange -> {{X[ϵ] - 2, 
            X[ϵ] + 30}, {L - 5 ϵ, L + 5 ϵ}}, 
      Epilog -> {{Green, 
            Line[{{X0, L - ϵ}, {1000, L - ϵ}}], 
            Line[{{X0, L + ϵ}, {1000, L + ϵ}}]}, {Red, 
            Line[{{X[ϵ], -100}, {X[ϵ], 100}}]}}, 
      Axes -> False, Frame -> True]

The above picture looks quite tight. It is difficult to say whether the blue line is below the green line. But, we can always choose X(ϵ) to be a large value:

Clear[ϵ, X, 
        X0] ;

L = 0;

X0 = 2;

X[ϵ_] := (2/(ϵ^2)) + 2

Now the picture is more convincing.

ϵ = 0.1;

Plot[{ff[x], L}, {x, X0, 1000}, PlotStyle -> {Blue, Black}, 
      PlotRange -> {{X[ϵ] - 2, 
            X[ϵ] + 30}, {L - 5 ϵ, L + 5 ϵ}}, 
      Epilog -> {{Green, 
            Line[{{X0, L - ϵ}, {1000, L - ϵ}}], 
            Line[{{X0, L + ϵ}, {1000, L + ϵ}}]}, {Red, 
            Line[{{X[ϵ], -100}, {X[ϵ], 100}}]}}, 
      Axes -> False, Frame -> True]

Even for a smaller ϵ

ϵ = 0.05;

Plot[{ff[x], L}, {x, X0, 1000}, PlotStyle -> {Blue, Black}, 
      PlotRange -> {{X[ϵ] - 2, 
            X[ϵ] + 30}, {L - 5 ϵ, L + 5 ϵ}}, 
      Epilog -> {{Green, 
            Line[{{X0, L - ϵ}, {1000, L - ϵ}}], 
            Line[{{X0, L + ϵ}, {1000, L + ϵ}}]}, {Red, 
            Line[{{X[ϵ], -100}, {X[ϵ], 100}}]}}, 
      Axes -> False, Frame -> True]

ϵ = 0.01;

Plot[{ff[x], L}, {x, X0, 100000}, PlotStyle -> {Blue, Black}, 
      PlotRange -> {{X[ϵ] - 2, 
            X[ϵ] + 30}, {L - 5 ϵ, L + 5 ϵ}}, 
      Epilog -> {{Green, 
            Line[{{X0, L - ϵ}, {100000, L - ϵ}}], 
            Line[{{X0, L + ϵ}, {100000, L + ϵ}}]}, {Red, 
            Line[{{X[ϵ], -100}, {X[ϵ], 100}}]}}, 
      Axes -> False, Frame -> True]

Example 3

The function that we are studing is from the notes.

ff[x_] := ((2  x)/(5 x  - 3))

We calculated in the notes

Clear[ϵ, X, X0] ;

    
L = 2/5;

X0 = 2;

X[ϵ_] := (1/ϵ) + 2

Now plot what we found. Remember that the blue graph must be between the green horizontal lines for all x > X(ϵ).  I put a vertical red line to mark X(ϵ),

But to get a picture we need to set a specific ϵ.

ϵ = 0.1;

    
Plot[{ff[x], L}, {x, X0, 1000}, PlotStyle -> {Blue, Black}, 
      PlotRange -> {{X[ϵ] - 2, 
            X[ϵ] + 30}, {L - 5 ϵ, L + 5 ϵ}}, 
      Epilog -> {{Green, 
            Line[{{X0, L - ϵ}, {1000, L - ϵ}}], 
            Line[{{X0, L + ϵ}, {1000, L + ϵ}}]}, {Red, 
            Line[{{X[ϵ], -100}, {X[ϵ], 100}}]}}, 
      Axes -> False, Frame -> True]

Example 4

The function that we are studing is from the notes.

Clear[ff];

ff[x_] := x - (Sqrt[(x^2) - x])

We calculated in class

Clear[ϵ, X, X0] ;

    
L = 1/2;

X0 = 2;

X[ϵ_] := Max[(1/(2 ϵ)), 2]

Now plot what we found. Remember that the blue graph must be between the green horizontal lines for all x > X(ϵ).  I put a vertical red line to mark X(ϵ),

But to get a picture we need to set a specific ϵ.

ϵ = 0.05;

Plot[{ff[x], L}, {x, X0, 1000}, PlotStyle -> {Blue, Black}, 
      PlotRange -> {{X[ϵ] - 2, 
            X[ϵ] + 30}, {L - 5 ϵ, L + 5 ϵ}}, 
      Epilog -> {{Green, 
            Line[{{X0, L - ϵ}, {1000, L - ϵ}}], 
            Line[{{X0, L + ϵ}, {1000, L + ϵ}}]}, {Red, 
            Line[{{X[ϵ], -100}, {X[ϵ], 100}}]}}, 
      Axes -> False, Frame -> True]