Limit of function as it approaches origin

Question

Find the limit of $f$ or prove that limit does not exist as $(x,y)$ approaches origin point.

$$f(x,y) = \frac{2x}{x^2 + x+ y^2}$$

So far, I have used several methods, such as finding correct $2$ paths for nonexistence of limit or applying polar coordinates, but I could not solve it properly. Any help would be appreciated!

What do you mean you could not solve it "properly"? What "wrong" answers did you get? — sammy gerbil, Apr 05 '20 at 15:32
Possible duplicate of Is there a step by step checklist to check if a multivariable limit exists and find its value? — sammy gerbil, Apr 05 '20 at 17:38

score -1 · Answer 1 · answered Apr 05 '20 at 16:45

-1

$$f(x,x)=\frac{2x}{2x^2+x}\rightarrow2.$$ $$f(x^2,x)=\frac{2x^2}{x^4+2x^2+x^2}\rightarrow\frac{2}{3},$$ which says that the limit does not exist.

answered Apr 05 '20 at 16:45

Michael Rozenberg

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Limit of function as it approaches origin

1 Answers1