Why does sin(xy)/x equal y as (x,y) approaches (0,y)?

[trap101]

So I figured out the solution to lim (x,y)-->(0,y) of Sin(xy)/x, but I figured it out by looking at a solution. I wanted to understand though why with respect to the above limit how Sin(xy)/x = y?

How do you separate the xy in the numerator?

 

[Ray Vickson]

So I figured out the solution to lim (x,y)-->(0,y) of Sin(xy)/x, but I figured it out by looking at a solution. I wanted to understand though why with respect to the above limit how Sin(xy)/x = y?

How do you separate the xy in the numerator?

Use l'Hospitals's rule, or else look at the behavior of $\sin \theta$ for small $|\theta|$.

 

[trap101]

Use l'Hospitals's rule, or else look at the behavior of $\sin \theta$ for small $|\theta|$.

Using L'Hospital's rule, would I essentially be treating "y" as a constant?

 

[Dick]

Using L'Hospital's rule, would I essentially be treating "y" as a constant?

Yes!