Help visualising a limit proof

[transgalactic]

i tried to interpret it on a sketch

http://img181.imageshack.us/img181/6498/13549423mx3.gif

 

[transgalactic]

i can't see the animation of this proove

 

[HallsofIvy]

When you don't know where to start, look at the definitions!

"\lim_{x\rightarrow x_0} f(x)= L" means, by definition, that

"Given any [\itex]\epsilon> 0[/itex] there exist \delta&gt; 0 such that if |x- x_0|&lt; \delta then [/itex]|f(x)- L|< \epsilon[/itex]".

Here, the function is just f(x)= x and you want to prove that the limit is x_0: write exactly the same thing but replace "f(x)" with "x" and "L" with "x_0".

"Given any [\itex]\epsilon> 0[/itex] there exist \delta&gt; 0 such that if |x- x_0|&lt; \delta then [/itex]|x- x_0|< \epsilon[/itex]".

You should see immediately that what was before "|f(x)- L|&lt; \epsilon" is now "|x- x_0|&lt; \epsilon" the same as with "|x- x_0|&lt; \delta". So make \epsilon and \delta the same: given any \delta&gt; 0, you can always choose \delta= \epsilon.

 

[HallsofIvy]

When you don't know where to start, look at the definitions!

"\lim_{x\rightarrow X} f(x)= L" means, by definition, that

"Given any [\itex]\epsilon> 0[/itex] there exist \delta&gt; 0 such that if |x- X|&lt; \delta then [/itex]|f(x)- L|< \epsilon[/itex]".

Here, the function is just f(x)= x and you want to prove that the limit is X: write exactly the same thing but replace "f(x)" with "x" and "L" with "X".

"Given any [\itex]\epsilon> 0[/itex] there exist \delta&gt; 0 such that if |x- X|&lt; \delta then [/itex]|x- X|< \epsilon[/itex]".

You should see immediately that what was before "|f(x)- L|&lt; \epsilon" is now "|x- X|&lt; \epsilon" the same as with "|x- X|&lt; \delta". So make \epsilon and \delta the same: given any \delta&gt; 0, you can always choose \delta= \epsilon.

As for your picture, it would be better to draw it in two dimensions: draw the graph of y= x, a straight line. Since [/itex]\epsilon[/itex] is bounding the value of the function, draw two horizontal lines at Y= X+ \epsilon and at y= X- \epsilon. Draw vertical lines where the graph crosses those horizontal lines to get a "box" bounding the graph. The vertical lines will give X+\delta and x-\delta. Here, of course, because the line is y= x, that box is a square: \delta= \epsilon.