Solving Limits with Delta-Epsilon Proofs

In summary, the author is trying to find the limit of a function in one variable, but is stuck because he is not sure how to do it. He converts to polar coordinates and finds that the limit is 2.
  • #1
ineedhelpnow
651
0
PLEASE HELP! i am so lost on this. we're using delta epsilon proofs and i am so confused since it was never properly taught to me in calc 1.

find the limit.
$\lim_{{(x,y)}\to{(0,0)}}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}$
 
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  • #2
I'm thinkin' you should convert to polar coordinates...

Usually you are given the value of the limit in an epsilon-delta proof. To simply find the value of this limit, convert to polars, then use L'Hôpital. What do you find?
 
  • #3
im confused on how to find the limit of this...
 
  • #4
ineedhelpnow said:
im confused on how to find the limit of this...

Converting to polar coordinates means to use:

\(\displaystyle r^2=x^2+y^2\)

Then you will have a limit in one variable...
 
  • #5
im lookin at the way my book would do it and first that set f(x,y)=(whatever your taking the limit of) and then they do f(x,0) (which I am stuck on) and f(0,y). by using polar coordinates, wouldn't i end up with a 0 in the denominator?
 
  • #6
ineedhelpnow said:
im lookin at the way my book would do it and first that set f(x,y)=(whatever your taking the limit of) and then they do f(x,0) (which I am stuck on) and f(0,y). by using polar coordinates, wouldn't i end up with a 0 in the denominator?

You will end up with the indeterminate form 0/0, and so this is why I suggested our friend L'Hôpital. :D
 
  • #7
oh i got it using another method. the answer is 2. is that the final answer though?

- - - Updated - - -

i think the way my book wants me to do it is by testing it with several functions to see if it all approaches the same limit.
 
  • #8
Yes, 2 is what I got...

\(\displaystyle \lim_{r\to0}\frac{r^2}{\sqrt{r^2+1}-1}=\lim_{r\to0}\frac{2r}{\dfrac{2r}{2\sqrt{r^2+1}}}=\lim_{r\to0}2\sqrt{r^2+1}=2\)

edit: I wanted to add that since when we converted to polar coordinates, and there was no $\theta$, we know we are approaching the limit from "all angles." :D
 

FAQ: Solving Limits with Delta-Epsilon Proofs

What is a delta-epsilon proof?

A delta-epsilon proof is a method for showing that a limit of a function exists by using the formal definition of a limit. It involves using the concepts of delta (change in x) and epsilon (change in y) to show that for any given epsilon, there exists a corresponding delta that satisfies the limit definition.

Why do we use delta-epsilon proofs?

Delta-epsilon proofs are used to rigorously prove the existence of a limit. This is important because it ensures that the limit is not just an assumption or an estimation, but a mathematically proven fact. It also allows for more precise and accurate calculations of limits.

What is the process for solving a limit with a delta-epsilon proof?

The process for solving a limit with a delta-epsilon proof involves setting up the limit definition, manipulating it to get an expression for delta in terms of epsilon, choosing a suitable value for delta, and showing that the limit definition is satisfied for that value of delta. This involves using algebraic and logical reasoning to make deductions and arrive at a proof.

What are the common challenges when using delta-epsilon proofs?

One of the common challenges when using delta-epsilon proofs is choosing a suitable value for delta. This value should satisfy the limit definition and also be small enough to accurately represent the change in x. Another challenge is understanding and applying the limit definition correctly, as it involves manipulating limits in a precise and logical manner.

Are there any alternatives to using delta-epsilon proofs for solving limits?

Yes, there are alternatives to using delta-epsilon proofs for solving limits. Other methods include using algebraic manipulations, graphing and visualizing the function, and using other limit theorems and properties. However, delta-epsilon proofs are often considered the most rigorous and precise method for proving the existence of a limit.

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