- #1

tmt1

- 234

- 0

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

I can set $y = mx$ to see if the function solely depends on $m$, in which case the limit does not exist. So I would get

$$\lim_{{(x, y)}\to{(0,0)}} \frac{x}{x^2 + m^2x^2}$$

Or$$\lim_{{(x, y)}\to{(0,0)}} \frac{x}{x^2(1 + m^2)}$$

Or

$$\lim_{{(x, y)}\to{(0,0)}} \frac{1}{x(1 + m^2)}$$

So in this case, the function does not depend on $m$, therefore this method is inconclusive. (Is it possible to use this method to evaluate a limit, or just to prove that it does not exist?).

I can try to use polar coordinates like this:

$$\lim_{{(x, y)}\to{(0,0)}} \frac{x}{x^2 + y^2} = \lim_{{r}\to{0}} \frac{r cos\theta}{r^2 (cos^2\theta + sin^2cos\theta)} = \lim_{{r}\to{0}} \frac{cos\theta}{r (cos^2\theta + sin^2cos\theta)}$$

I'm not sure how to simplify from here, so it appears this method is inconclusive.

Or, I can try to evaluate the different limits along the x and y axes.

So the limit along the x-axis would be

$$\lim_{{x}\to{(0)}} \frac{x}{x^2} = \infty$$And the limit along the y-axis would be

$$\lim_{{y}\to{(0)}} \frac{0}{y^2} = 0$$

Therefore, the limit does not exist, as the limits on both axes are different.

Were these methods applied correctly?