How to solve this limit? limit, (x,y)=>(0,0), of sin(xy) / (xy)

[Anto]

Hello, can someone help me with this exercise please ?

I have to study this limit when x and y tends to (0,0) \(\displaystyle f(x,y) = sin(xy)/(xy) \)

Aproximating the limit I get that it exists. However, I don't know how to calculate it.

 

[stapel]

Hello, can someone help me with this exercise please ?

I have to study this limit when x and y tends to (0,0) \(\displaystyle f(x,y) = sin(xy)/(xy) \)

Aproximating the limit I get that it exists. However, I don't know how to calculate it.

Are you able to refer at all to the basic trig limit you learned back in first-semester calculus? Namely:

. . . . .\(\displaystyle \displaystyle \lim_{\theta\, \rightarrow\, 0}\, \dfrac{\sin(\theta)}{\theta}\, =\, 1\)

I think this is sufficient (rename "xy" as "z", say, and let z goes to zero), but you can review another (much more advanced) argument here.