PlanetMath: iterated limit in $\mathbb{R}^2$

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

(Definition)

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

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

are called iterated limits.

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

$\lim_{x\to 0}\left(\lim_{y\to 0}f(x,\,y)\right) = \lim_{x\to0}\sin\frac{1}{x}$ does not exist
$\lim_{y\to 0}\left(\lim_{x\to 0}f(x,\,y)\right) = \lim_{y\to0}1 = 1$
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

$\lim_{x\to 0}\left(\lim_{y\to 0}f(x,\,y)\right) = \lim_{x\to0} \frac{x^2}{x^2} = 1$
$\lim_{y\to 0}\left(\lim_{x\to 0}f(x,\,y)\right) = \lim_{y\to0} 0 = 0$
the usual limit $\lim_{(x,y)\to(0,0)}f(x,\,y)$ again does not exist, even 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

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

then (we apply l'Hôpital's rule throughout)

$\lim_{x\to 0}\left(\lim_{y\to 0}f(x,\,y)\right) = \lim_{x\to 0}\left(\lim_{y\t... ...{x\to 0}\frac{x\sin x}{x^2}=\lim_{x\to 0}\frac{\sin x}{x}=\lim_{x\to 0}\cos x=1$
$\lim_{y\to 0}\left(\lim_{x\to 0}f(x,\,y)\right) = \lim_{y\to 0}\left(\lim_{x\t... ...to 0}\frac{y\sinh y}{y^2}=\lim_{y\to 0}\frac{\sinh y}{y}=\lim_{y\to 0}\cosh y=1$
the usual limit $\lim_{(x,y)\to(0,0)}f(x,\,y)$ exists in this case. An essential reason which assures the continuity of this function, arises from the fact that $f(x,\,y) \equiv \Re(\frac{\sin z}{z})$ , , i.e. it is the real part of the analytic function $w := \frac{\sin z}{z}$ having the removable singularity at (see the entry complex sine and cosine).

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Cross-references: complex sine and cosine, removable singularity, analytic function, real part, point, limits, accumulation point, subset, function
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This is version 7 of iterated limit in $\mathbb{R}^2$ , born on 2007-08-13, modified 2007-08-19.
Object id is 9860, canonical name is IteratedLimitInMathbbR2.
Accessed 981 times total.

Classification:

AMS MSC:	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

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