- #1
AxiomOfChoice
- 533
- 1
I'm trying to show the following:
[tex]
\lim_{(x,y) \to (0,0)} \frac{x^2 + \sin^2 y}{x^2 + y^2} = 1.
[/tex]
One can show that
[tex]
\frac{x^2 + \sin^2 y}{x^2 + y^2} \leq 1
[/tex]
for all [itex]x,y[/itex] because [itex]\sin y \leq y[/itex]. So, if you can bound this guy from below by something that goes to 1 as [itex](x,y) \to (0,0)[/itex], you should be in business by the Sandwich Theorem. But I have so far been unable to do that! Does anyone have any suggestions as to how to proceed?
[tex]
\lim_{(x,y) \to (0,0)} \frac{x^2 + \sin^2 y}{x^2 + y^2} = 1.
[/tex]
One can show that
[tex]
\frac{x^2 + \sin^2 y}{x^2 + y^2} \leq 1
[/tex]
for all [itex]x,y[/itex] because [itex]\sin y \leq y[/itex]. So, if you can bound this guy from below by something that goes to 1 as [itex](x,y) \to (0,0)[/itex], you should be in business by the Sandwich Theorem. But I have so far been unable to do that! Does anyone have any suggestions as to how to proceed?