# Taking the limit of this multiple variable function

#### Monkens

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)

#### Mush

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
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)
$$\displaystyle x \to ? ,y \to ?$$

#### Monkens

I am very sorry, it's at the origin (0,0)

#### Laurent

MHF Hall of Honor
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.

#### Monkens

Thank you very much, that's really helpful I never thought of substituting u in then using the taylor polynomial for that.