Thread: A multivariable limit I need to evaluate

1. A multivariable limit I need to evaluate

Do I have to use the precise definition of a limit to evaluate the following?

lim (x^2+y^2)/((root(x^2+y^2+1) - 1)
(x,y)-->(0,0)

I read through the section in the book, and the answer the book gives for this limit is 2. I don't know how to attack this limit.

Thanks for the help.

2. Originally Posted by Undefdisfigure
Do I have to use the precise definition of a limit to evaluate the following?

lim (x^2+y^2)/((root(x^2+y^2+1) - 1)
(x,y)-->(0,0)

I read through the section in the book, and the answer the book gives for this limit is 2. I don't know how to attack this limit.

Thanks for the help.
to use the precise definition of the limit (and i am not totally sure what you are referring to when you say this, but) you have to have an idea of what the limit is to begin with. so you approach the limit along several lines, and you should always get two, so you would believe 2 is the limit and then try to prove it by the definition of the limit.

here's another way though: switch to polar coordinates

recall that, in polar coordinates:

$\displaystyle r^2 = x^2 + y^2$

so as $\displaystyle (x,y) \to (0,0)$, we have $\displaystyle r \to 0$

thus,

$\displaystyle \lim_{(x,y) \to (0,0)} \frac {x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} = \lim_{r \to 0} \frac {r^2}{\sqrt{r^2 + 1} - 1}$

so we used a change of variable to change to a limit in one variable, i'm sure you can take it from here

3. Originally Posted by Undefdisfigure
Do I have to use the precise definition of a limit to evaluate the following?

lim (x^2+y^2)/((root(x^2+y^2+1) - 1)
(x,y)-->(0,0)

I read through the section in the book, and the answer the book gives for this limit is 2. I don't know how to attack this limit.

Thanks for the help.
$\displaystyle \frac{x^2+y^2}{\sqrt{x^2+y^2+1} - 1} \cdot \frac{\sqrt{x^2+y^2+1} + 1}{\sqrt{x^2+y^2+1} + 1}$
We get,
$\displaystyle \frac{(x^2+y^2)(\sqrt{x^2+y^2+1} + 1)}{(x^2+y^2+1)-1}$
It is easy to see the limit is 2.