Does the limit of $\frac{x}{y}$ depend on the direction we approach from?

Question

Consider $f(x,y) = \frac{x}{y}$

Suppose we want to consider $$ \lim_{x\to 0,y\to 0} \frac{x}{y} $$

Does this limit exist if we approach from different directions?

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I figure if we approach from the horizontal axis, then for any $x$ the function will be very large.

However, if we approach such that $\frac{x}{y}=1$ at every point (i.e. if we approach along the line $y=x$ then shouldn't this limit exist (and $=1$??))

If that is true though... then wouldn't there be infinitely many limits depending on the line along which we approach (i.e. if we approach from $y=2x$, then the limit would be $.5$ etc)

Answer 1

Using polar coordinates,

$$\lim_{\rho\to0}\frac{\rho\cos\theta}{\rho\sin\theta}=\lim_{\rho\to0}\frac{\cos\theta}{\sin\theta}=\cot\theta.$$

This gives the value of the limit-when-coming-in-the-direction-$\theta$. (It does not exist when coming along the axis $x$, where $\theta=k\pi$.)

At the same time, this shows that

$$\lim_{x,y\to0}\frac xy$$ does not exist.

Answer 2

You are essentially correct, but the terminology is that "the limit does not exist".

A limit is defined to exist if and only if the limit is the same from all directions. Proving that a limit does not exist requires only showing that if approached from 2 different directions it is not the same.