- #1
ImAnEngineer
- 209
- 1
Homework Statement
Compute:
1)[tex]lim_{(x,y)\rightarrow (0,0)} \frac{x^3-y^3}{x^2+y^2}[/tex]
2)[tex]lim_{(x,y)\rightarrow (0,0)} \frac{sin(xy)}{y}[/tex]
The Attempt at a Solution
1)
[tex]lim_{(x,y)\rightarrow (0,0)} \frac{x^3-y^3}{x^2+y^2}[/tex]
Not sure what to do here. Either the limit doesn't exist or it equals zero. I think it equals zero because in the special case where x=y we get 0/2x²=0. Also when x=0, we get the limit of -y where y approaches zero, which is zero. Same goes for y=0. But I don't know how to prove that the limit is zero in general.
2)
[tex]lim_{(x,y)\rightarrow (0,0)} \frac{sin(xy)}{y}=lim_{z\rightarrow 0} \frac{sin(z)}{\frac{z}{x}}=lim_{z\rightarrow 0} x\frac{sin(z)}{z}=lim_{z\rightarrow 0}x lim_{z\rightarrow 0} \frac{sin(z)}{z}=0*x=0
[/tex]
Are all steps correct? I'm not sure about the substitution and using the product rule for limits.