Q:

when computing $\;\lim_{x\to a, y\to b} f(x,y) $, why is it there are infinite directions of letting (x,y) approach (a,b)? 

Suppose we are trying to compute $$\;\lim_{x\to a, y\to b} f(x,y) $$
My textbook claims 

For functions of a single variable, when we let x approach a, there
  are only two possible directions of approach, from the left or from
  the right...For functions of two variables the situation is not as
  simple because we can let (x,y) approach (a,b) from an infinite number
  of directions in any manner whatsoever (see figure below) as long as
  (x,y) stays within the domain of f.

My question: why infinite?  If I remember calculus 1 correctly, when taking limits of a single variable $$\;\lim_{x\to a} f(x) $$ you approach $a$ when $x < a$ and see where it takes you.   Then you approach $a$ when $x > a$. That adds to two ways of getting to $a$. In the multivariable case you do the exact same thing with an additional two steps: approach $b$ when $y<b$ and approach $b$ when $y>b$. That's a total of 4 ways to approach $(a,b)$ So how are we getting to infinite ways?  

A:

For instance, you can approach $(a,b)$ along a straight line of any slope, which is already an infinite number of ways. 

A:

When talking about limits in a multivariate environment forget about "directions" when approaching a point ${\bf p}=(a,b)$. Note that some advocatus diaboli could even force you to approach ${\bf p}$ along some spiral!  Therefore the "infinity" you have to face when working with this path concept for limits is much larger than you thought. 
Note that limits are defined in terms of distances, $\epsilon$, and $\delta$. The fact
$$\lim_{(x,y)\to(a,b)} f(x,y)=z$$
holds if for any given $\epsilon>0$ you can find a $\delta>0$ such that
$$0<\|(x,y)-(a,b)\|<\delta\quad\Rightarrow\quad |f(x,y)-z|<\epsilon\ .$$
Here the distance $\|(x,y)-(a,b)\|$ could be euclidean distance $\sqrt{(x-a)^2+(y-b)^2}$, but one of
$$|x-a|+|y-b],\qquad \max\{|x-a|, |y-b|\}$$
would serve as well.  The former of these two is called $l^1$ distance, the latter $l^\infty$ distance. 
By the way: You wrote $\lim_{x\to a, \ y\to b} f(x,y)$. The limit described in the above has to be distinguished from the "iterated limit" $\lim_{x\to a}\>\lim_{y\to b} f(x,y)$.

