Thread: Two-Variable Limit and Continuity

I'm having trouble with this problem

f(x,y)= {(sin(x^2+y^2 )/((x^2+y^2 ) ) (x,y)=(0,0)
1 (x,y)≠(0,0)


The question says to show that f is continuous

Originally Posted by ghow90

I'm having trouble with this problem

f(x,y)= {(sin(x^2+y^2 )/((x^2+y^2 ) ) (x,y)=(0,0)
1 (x,y)≠(0,0)


The question says to show that f is continuous

I assume that you actually mean

for

and for .


To show that this is continuous, you'd have to show that

.






since .



It is well known that (and can be shown using the sandwich theorem).


Therefore .


Since this limit is , the function is continuous.

Originally Posted by Prove It

I assume that you actually mean

for

and for .


To show that this is continuous, you'd have to show that

.






since .



It is well known that (and can be shown using the sandwich theorem).


Therefore .


Since this limit is , the function is continuous.

Thank You for the help

Notice that Prove It changed to polar coordinates. That simplifies the problem greatly- since just r itself measures the distance to (0,0) regardless of . As long as the limit, as r goes to 0, does NOT depend on [tex]\theta[\math], that gives the correct limit.