How do I find the limit of #(xy)/sqrt (x^2+y^2)#?
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...
if we change the coordinate system from cartesian to polar, we will have two new variables
If the limit will depend only from
Now let's do the exercise.
I remember that:
and, it's easy to say, this limit doesn't depends from
You can tell the limit is going to be zero by comparing the degrees of the top and bottom.
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