Iterated limit

In multivariable calculus, an iterated limit is an expression of the form

\lim _{y\to q}{\big (}\lim _{x\to p}f(x,y){\big )}.\,

One has an expression whose value depends on at least two variables, one takes the limit as one of the two variables approaches some number, getting an expression whose value depends only on the other variable, and then one takes the limit as the other variable approaches some number. This is not defined in the same way as the limit

\lim _{(x,y)\to (p,q)}f(x,y),\,

which is not an iterated limit. To say that this latter limit of a function of more than one variable is equal to a particular number L means that ƒ(x, y) can be made as close to L as desired by making the point (x, y) close enough to the point (p, q). It does not involve first taking one limit and then another.

Counterexamples

It is not in all cases true that

\lim _{(x,y)\to (p,q)}f(x,y)=\lim _{x\to p}\lim _{y\to q}f(x,y)=\lim _{y\to q}\lim _{x\to p}f(x,y).

(1)

Among the standard counterexamples are those in which

f(x,y)={\frac {x^{2}}{x^{2}+y^{2}}}

and

f(x,y)={\frac {xy}{x^{2}+y^{2}}},

[1]

and (p, q) = (0, 0).

In the first example, the values of the two iterated limits differ from each other:

\lim _{y\to 0}\left(\lim _{x\to 0}{\frac {x^{2}}{x^{2}+y^{2}}}\right)=\lim _{y\to 0}0=0,

and

\lim _{x\to 0}\left(\lim _{y\to 0}{\frac {x^{2}}{x^{2}+y^{2}}}\right)=\lim _{x\to 0}1=1.

[2]

In the second example, the two iterated limits are equal to each other despite the fact that the limit as (x, y) → (0, 0) does not exist:

\lim _{x\to 0}\left(\lim _{y\to 0}{\frac {xy}{x^{2}+y^{2}}}\right)=\lim _{x\to 0}0=0

and

\lim _{y\to 0}\left(\lim _{x\to 0}{\frac {xy}{x^{2}+y^{2}}}\right)=\lim _{y\to 0}0=0,

but the limit as (x, y) → (0, 0) along the line y = x is different:

\lim _{{\Big (}(x,y)\to (0,0)\,:\,y=x{\Big )}}{\frac {xy}{x^{2}+y^{2}}}=\lim _{x\to 0}{\frac {x^{2}}{x^{2}+x^{2}}}={\frac {1}{2}}.

It follows that

\lim _{(x,y)\to (0,0)}{\frac {xy}{x^{2}+y^{2}}}

does not exist.

Sufficient condition

A sufficient condition for (1) to hold is Moore-Osgood theorem: If $\lim _{x\to p}f(x,y)$ exists pointwise for each y different from q and if $\lim _{y\to q}f(x,y)$ converges uniformly for x≠p then the double limit and the iterated limits exist and are equal.[3]

References

Stewart, James (2008). "Chapter 15.2 Limits and Continuity". Multivariable Calculus (6th ed.). pp. 907–909. ISBN 0495011630.
Although this is not wrong, you should pay attention the fact
$\lim _{y\to 0}{\frac {x^{2}}{x^{2}+y^{2}}}={\begin{cases}1&{\text{for }}x\neq 0\\0&{\text{for }}x=0\end{cases}}$ .
(But this is a minor problem since we will soon think under $\lim _{x\to 0}$ .)
Taylor, Angus E. (2012). General Theory of Functions and Integration. Dover Books on Mathematics Series. p. 140. ISBN 9780486152141.

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[1] Stewart, James (2008). "Chapter 15.2 Limits and Continuity". Multivariable Calculus (6th ed.). pp. 907–909. ISBN 0495011630.

[2] Although this is not wrong, you should pay attention the fact
$\lim _{y\to 0}{\frac {x^{2}}{x^{2}+y^{2}}}={\begin{cases}1&{\text{for }}x\neq 0\\0&{\text{for }}x=0\end{cases}}$ .
(But this is a minor problem since we will soon think under $\lim _{x\to 0}$ .)

[3] Taylor, Angus E. (2012). General Theory of Functions and Integration. Dover Books on Mathematics Series. p. 140. ISBN 9780486152141.

Iterated limit

Counterexamples

Sufficient condition

See also

References