# iterated limit in $\mathbb{R}^{2}$

Let $f$ be a function  from a subset $S$ of  $\mathbb{R}^{2}$  to  $\mathbb{R}$ and  $(a,\,b)$  an accumulation point   of $S$. The limits

 $\lim_{x\to a}\left(\lim_{y\to b}f(x,\,y)\right)\quad\mbox{and}\quad\lim_{y\to b% }\left(\lim_{x\to a}f(x,\,y)\right)$

are called iterated limits.

Example 1. If  $\displaystyle f(x,\,y):=\frac{x\sin\frac{1}{x}+y}{x+y}$,  then

• $\lim_{x\to 0}\left(\lim_{y\to 0}f(x,\,y)\right)=\lim_{x\to 0}\sin\frac{1}{x}$ does not exist

• $\lim_{y\to 0}\left(\lim_{x\to 0}f(x,\,y)\right)=\lim_{y\to 0}1=1$

• the usual limit $\lim_{(x,y)\to(0,0)}f(x,\,y)$ does not exist.

Example 2. If  $\displaystyle f(x,\,y):=\frac{x^{2}}{x^{2}+y^{2}}$,  then

• $\lim_{x\to 0}\left(\lim_{y\to 0}f(x,\,y)\right)=\lim_{x\to 0}\frac{x^{2}}{x^{2% }}=1$

• $\lim_{y\to 0}\left(\lim_{x\to 0}f(x,\,y)\right)=\lim_{y\to 0}0=0$

• the usual limit $\lim_{(x,y)\to(0,0)}f(x,\,y)$ again does not exist, though both of the iterated limits do.

So far we have studied examples that present discontinuity at its point of accumulation. We now expose an illustrative example where such discontinuity can be avoided.

Example 3. Consider the function

 $f(x,\,y):=\frac{x\sin{x}\cosh{y}+y\cos{x}\sinh{y}}{x^{2}+y^{2}};$

then (we apply l’Hôpital’s rule (http://planetmath.org/LHpitalsRule) throughout)

Title iterated limit in $\mathbb{R}^{2}$ IteratedLimitInmathbbR2 2013-03-22 17:28:27 2013-03-22 17:28:27 pahio (2872) pahio (2872) 10 pahio (2872) Definition msc 26B12 msc 26A06 iterated limit