Multivariable limit... can I assure it does not exist? Should I continue simplifying?

patr1c1a

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I'm trying to analyze continuity for the following function at (0,0):

attachment.php


I tried a few approximations that I managed to solve to 0. But then I tried this approximation when x0 ; y=x2 and this is how far I got:
attachment.php

I'm not sure if there is nothing else I can do to simplify so I must conclude the limit doesn't exist, or if I'm actually doing something wrong and the limit exists (and it's 0). So I tried compression:
attachment.php

but again, I'm stuck there as I can't find the g(x,y) function I need...

Any help would be much appreciated :)
Thanks in advance!
 

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Why would you approach (0, 0) along the path \(\displaystyle y= x^2\)? Since x and y appear in exactly the same way, it would make more sense to me to try y= x.

Along the line y= x, we are looking at \(\displaystyle \lim_{x\to 0}\frac{x^4+ sin(x^4)}{2x^2}= \lim_{x\to 0} \frac{1}{2}x^2+ \frac{sin(x^4}{2x^2}\).

Now, it should be obvious that \(\displaystyle \frac{sin(x^4)}{2x^2}\), since the numerator lies between -1 and 1 while the denominator goes to infinity, will go to 0 but that leaves \(\displaystyle \frac{1}{2}x^2\).
 
I'm trying to analyze continuity for the following function at (0,0):

attachment.php


I tried a few approximations that I managed to solve to 0. But then I tried this approximation when x0 ; y=x2 and this is how far I got:
attachment.php

I'm not sure if there is nothing else I can do to simplify so I must conclude the limit doesn't exist, or if I'm actually doing something wrong and the limit exists (and it's 0). So I tried compression:
attachment.php

but again, I'm stuck there as I can't find the g(x,y) function I need...

Any help would be much appreciated :)
Thanks in advance!
Your mistake is following the equals sign below
\(\displaystyle \dfrac{x^4\, + sin(x^8)}{x^2\, +\, x^4}\, \dfrac{x^6}{x^6}\, =\, ...\),
i.e. you do not get a term like
\(\displaystyle \dfrac{1}{x^4(1+x^2)}\)

What you might do is write the numerator as
\(\displaystyle x^4\, \large (1 + \dfrac{sin(x^8)}{x^4}\large )\)
=\(\displaystyle \, x^4\, \large (1 + \dfrac{x^4}{x^4}\, \dfrac{sin(x^8)}{x^4}\large )\)
=\(\displaystyle \, x^4\, \large (1 + x^4\, \dfrac{sin(x^8)}{x^8}\large )\)
=\(\displaystyle \, x^4\, \large (1 + x^4\, sinc(x^8)\large )\)

Whether you have sinc(x) or sinc(xn), both go to one as x goes to zero [ let u = xn and, since u goes to zero as x goes to zero, sinc(u) goes to one as u goes to zero.
 
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