# How do I find the limit of #(xy)/sqrt (x^2+y^2)#?

##### 2 Answers

I suppose that the limit is:

The answer is:

This limit is in the indecision form:

The limits of a function of more than one variable are really different from the ones of one variable. The variable

The way to demonstrate, and to calculate, a limit making **only one** limit is to make a calculation for every direction.

It seems to be not so easy...

**but**

if we change the coordinate system from cartesian to polar, we will have two new variables **all** the directions in one shot!

If the limit will depend **only** from

Now let's do the exercise.

I remember that:

and if

than

So:

and, it's easy to say, this limit **doesn't** depends from

#### Answer:

You can tell the limit is going to be zero by comparing the degrees of the top and bottom.

#### Explanation:

The numerator has "2nd order smallness" - it's "tiny" - because of the

The denominator has only "1st order of smallness" - it's just "small" - because the

So the quotient will be "tiny/small" which goes to 0.

"Number sense" - If x and y are around 1/100, the top will be about 1/10000, and