# What is being Done in This proof of Limits?

I have been trying to learn calculus by my own, but when it comes to proving limits I get very confuse.

Could somebody explain me what is being done here?   If you know any resources that could help me with this task let me know.

here is the source:
http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

Last edited:

mfb
Mentor
The first line is the statement you want to show.
The second line is a clever guess for delta (as function of epsilon), and the remaining steps are just simplifications, showing that |(5x-4)-6| is indeed < epsilon if |x-2|<delta.

Mark44
Mentor
I have been trying to learn calculus by my own, but when it comes to proving limits I get very confuse.

Could somebody explain me what is being done here?   If you know any resources that could help me with this task let me know.

here is the source:
http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

They're starting with this inequality:
##|(5x - 4) - 6| < \epsilon##
In a few algebra operations, they arrive at this:
##5|x - 2| < \epsilon ##
or
##|x - 2| < \epsilon/5 ##
If you let ##\delta = \epsilon/5##, then by reversing the steps above, you'll get back to the first inequality.

The whole idea is sort of a challenge-response. If you're trying to convince someone that ##\lim_{x \to 2}5x -4 = 6##, they might ask you get a function value within 0.1 (that's the ##\epsilon##). You say, take any x within 0.1/5 = 0.02 of 2.

If the challenger isn't satisfied, he might ask if you can get the function value within 0.001. You tell him to take any x within 0.0002 of 2 (i.e., between 1.9998 and 2.0002).

And so on. Eventually, he'll give up and accept that the limit is indeed 2.

Last edited:
They're starting with this inequality:
##|(5x - 4) - 6| < \epsilon##
In a few algebra operations, they arrive at this:
##5|x - 2| < \epsilon ##
or
##|x - 2| < \epsilon/5 ##
If you let ##\delta = \epsilon/5##, then by reversing the steps above, you'll get back to the first inequality.

The whole idea is sort of a challenge-response. If you're trying to convince someone that ##\lim_{x \to 2}5x -4 = 6##, they might ask you get a function value within 0.1 (that's the ##\epsilon##). You say, take any x within 0.1/5 = 0.02 of 2.

If the challenger isn't satisfied, he might ask if you can get the function value within 0.001. You tell him to take any x within 0.0002 of 2 (i.e., between 1.9998 and 2.0002).

And so on. Eventually, he'll give up and accept that the limit is indeed 2.
thanks

Last edited by a moderator: