- #1

- 14

- 0

I'm having a lot of difficulty with finding limits of multivariable functions. A question like this comes up every year in the final exam and it will always ask for use of the squeezing theorem.

## Homework Statement

(a) Suppose that

f(x, y) = 1 +(5x2y3)/x2 + y2

for (x, y) =/= (0, 0)

and that f(0, 0) = 0. By applying the Squeezing Rule to |f(x, y) − 1|, or otherwise, prove

that f(x, y) -> 1 as (x, y) -> (0, 0).

## Homework Equations

## The Attempt at a Solution

I understand that in order for a limit to exist that no matter what direction we approach (0,0) we must compute the same value. From x-axis and y-axis it seems that the limit is indeed 1. I also get the intuition of squeeze theorem that

lim (x,y) -> (a,b) g(x) <= lim (x,y) -> (a,b) f(x) <= lim (x,y) -> (a,b) h(x)

so lim g(x) = lim h(x) then we have found our lim f(x)

What I'm really confused about is how we set up the squeeze inequality that I see in some textbooks.

Would it be something like this ?

1 =< (5x2y3)/(x2 + y2) <= (I have no idea how you would find an expression on the RHS)