# Using delta epsilon. inequality manipulation

## Main Question or Discussion Point

Im having trouble proving limits of multivariable functions. I understand the principal behind delta-epsilon proofs but I cant get it to work. Once I set up the inequalities Im stuck.

The only example in my book seems very convenient though .

3x^2 * abs(y) divided by x^2 + y^2 It uses x^2 divided by x^2 +y^2 is less than or equal to one so (3x^2 * abs(y) divided by x^2 + y^2) is less than or equal to 3 abs(y).

Im guessing that you are supposed to manipulate equalitys to find delta in terms of epsilon in all cases, is that right?

It may be that I dont have the intuitive sense to just synthesize the proper way to do this "manipulation" in my homework problems. I understood the method the book used but it would not have been clear had it not been shown step by step. Id like to find some sources that would help me get a better feel for this process so I came here asking for help.

Also as someone who is looking forward to a carrer in math/physics how concerned should I be that I am unable to just DO this? The book assumes that I can get delta in terms of epsilon. I cant so I am concerned.

Im having trouble proving limits of multivariable functions. I understand the principal behind delta-epsilon proofs but I cant get it to work. Once I set up the inequalities Im stuck.

The only example in my book seems very convenient though .

3x^2 * abs(y) divided by x^2 + y^2 It uses x^2 divided by x^2 +y^2 is less than or equal to one so (3x^2 * abs(y) divided by x^2 + y^2) is less than or equal to 3 abs(y).

Im guessing that you are supposed to manipulate equalitys to find delta in terms of epsilon in all cases, is that right?

It may be that I dont have the intuitive sense to just synthesize the proper way to do this "manipulation" in my homework problems. I understood the method the book used but it would not have been clear had it not been shown step by step. Id like to find some sources that would help me get a better feel for this process so I came here asking for help.

Also as someone who is looking forward to a carrer in math/physics how concerned should I be that I am unable to just DO this? The book assumes that I can get delta in terms of epsilon. I cant so I am concerned.
yaje:

Before you get involved with multi variable limits you should check yourself how good you are with a single variable limits.

Can you for example do the following limits quite easily:

1) $$lim_{x\to 1}{2x+3} = 5$$

2)$$lim_{x\to 1}x^2 = 1$$

3) $$lim_{x\to 2}\sqrt x =\sqrt 2$$

4)$$lim_{x\rightarrow 0} sinx = 0$$