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I have an examples book with a limit exercise that I can't understand.

The limit in question is:

$$f(x,y)=\frac{x}{x+y}$$ with $x\ne-y$; $$\lim_{(x,y)\to(0,0)} f(x,y)$$

And then to solve it, it goes:

$$\lim_{(x,y)\to(0,0)} f(x,y) = \lim_{x\to0} f(x,mx) =\lim_{x\to 0}\frac{x}{x+mx}=\frac{1}{1+m}.$$

Can you help me understand that? Thanks,


UPDATE: Ok, just to make sure that I got it right. I have a very similar test exercise with $4$ different options.

The following limit $$\lim_{(x,y)\to(0,0)}\frac{-x^3+3xy^2}{x^2+y^2}$$ equals:

A. $0$

B. $- \infty$

C. Doesn't exist

D. $ +\infty$

My doubt is: if I consider it normally I'd say that it doesn't exist, but if I solve it using the same approach (i.e. $y=mx$) then the limit equals $0$. Which one is the right answer?

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    I think you are missing something important regarding $y$. Is $y$ a straight line massing through the origin..or anything like this has been mentioned in the book.2010-08-18
  • 1
    @Chandru1: Yes, you're right. Thanks for the correction. It says that "y is a straight line with any value for m but -1"2010-08-18

6 Answers 6