- #1

Puchinita5

- 183

- 0

## Homework Statement

Hi. I'm going to provide two homework examples. The first one I got correctly. But the second one, I used the same method as the first question but I got the answer incorrect. I'm really confused.

The first question was

[tex]

\lim_{(x,y)\to(0,0)}\frac{(xycosy)}{3x^2+y^2}

[/tex]

f(x,0)=0

f(0,y)=0

for y=x , f(x,x)= (1/4)cosx

since the three limits are not equal, the limit does not exist!

Okay now the second question:

[tex]

\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}

[/tex]

I again did:

f(x,0)=0

f(0,y)=0

[tex]f(x,x)=\frac{x^2}{x\sqrt{2}}[/tex]

since they aren't the same, limit does not exist. I THOUGHT.

I looked in the solutions manual, and it says "We can see that the limit along any line through (0,0) is 0, as well as along other paths through (0,0) such as x=y^2. So we suspect that the limit exists and equals 0; we use the Squeeze Theorem to prove our assertion.

[tex]

0\leq\left | (\frac{xy}{\sqrt{x^2+y^2}})\right |\leq \left |x \right | since \left |y\right |\leq \sqrt(x^2+y^2) and \left |x\right | \to \0 \ as (x,y) \to \(0,0).\ so\ the\ limit\ equals\ 0. [/tex]

I guess I don't understand how you choose the method to solve for the limit. Is there something about the function that tells you what to do? Why is f(x,x) not an option?

I mean, I guess it makes sense because I'm sure you can't just plug in anything or at some point you will get something that won't equal the same thing. But how do you know what to do? How would I have known to use the squeeze theorem?

I'm so confused.

[