Determine if the limit exists.

[mapabu]

I have this limit and need to know if it exists. I've tried different paths (already evaluated f(x,0),f(0,y),f(x,x²) and all of those paths have the same limit value: 0).That's why i suspect that the limit exists and it's value is 0. Then i tried to prove it using the Squeeze Theorem,but i need an inequality for eliminating the denominator and can't find which one . am i going the right way or is there a path with a different limit value? if there isn't, which inequality should i use? . hope you to help me out please! :wink:

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[daon2]

Sorry, I edited this post seeing that you already know the basics.

Hint: If \(\displaystyle a,b,c>0\) then \(\displaystyle \dfrac{a}{b+c} < \dfrac{a}{b}\).

Then use |f|->0 implies f->0.

 

[Bob Brown MSEE]

Squeeze Theorem?

View attachment 2816​

If sin(1/x) were involved then squeeze theorem would certainly be involved.
However, near the origin this function becomes ...

g(x) = xyxx/(xxxx+yy)

What would you use for g(x)?
Daon's post is on point.