Taking the limit of this multiple variable function

May 2010
23
1
Hey, hope this is in the right topic

\(\displaystyle f(x,y) = (sin(xy)-(xy))/sin(x^2+y^2)\)

Thats whats troubling me, I just really dont know where to start and I cant work it out from the solutions.


Thanks guys.(And gals)
 
Dec 2008
901
339
Scotland
Hey, hope this is in the right topic

\(\displaystyle f(x,y) = (sin(xy)-(xy))/sin(x^2+y^2)\)

Thats whats troubling me, I just really dont know where to start and I cant work it out from the solutions.


Thanks guys.(And gals)
Which limit are you trying to take? The limit as what tends to what?
 

pickslides

MHF Helper
Sep 2008
5,237
1,625
Melbourne
Hey, hope this is in the right topic

\(\displaystyle f(x,y) = (sin(xy)-(xy))/sin(x^2+y^2)\)

Thats whats troubling me, I just really dont know where to start and I cant work it out from the solutions.


Thanks guys.(And gals)
\(\displaystyle
x \to ? ,y \to ?
\)
 
May 2010
23
1
I am very sorry, it's at the origin (0,0)
 

Laurent

MHF Hall of Honor
Aug 2008
1,174
769
Paris, France
Hey, hope this is in the right topic

\(\displaystyle f(x,y) = (sin(xy)-(xy))/sin(x^2+y^2)\)

Thats whats troubling me, I just really dont know where to start and I cant work it out from the solutions.


Thanks guys.(And gals)
Let's prove the limit is 0. We know that \(\displaystyle \sin(u)-u\sim-\frac{u^3}{6}\) when \(\displaystyle u\to0\). Since \(\displaystyle xy\to 0\) when \(\displaystyle (x,y)\to(0,0)\), we have \(\displaystyle \sin(xy)-xy\sim-\frac{(xy)^3}{6}\) when \(\displaystyle (x,y)\to(0,0)\). Similarly, \(\displaystyle \sin(x^2+y^2)\sim x^2+y^2\), so that \(\displaystyle f(x,y)\sim-\frac{(xy)^3}{6(x^2+y^2)}\) when \(\displaystyle (x,y)\to(0,0)\).

Now we can say for instance \(\displaystyle \frac{|xy|^3}{x^2+y^2}\leq |xy^3|\frac{x^2}{x^2+y^2}\leq |xy^3|\) (since the ratio is less than 1) hence \(\displaystyle \frac{(xy)^3}{6(x^2+y^2)}\to0\)when \(\displaystyle (x,y)\to(0,0)\). This concludes.
 
May 2010
23
1
Thank you very much, that's really helpful :) I never thought of substituting u in then using the taylor polynomial for that.