Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Show that for f=x yx2+y2+x sin1/y the bivariate limit at the origin and the iterated limit limx→0limy→0f both fail to exist, but the iterated limit limy→0limx→0f is zero.
Since fx,m x=m1+m2+x sin1m x, the bivariate limit at the origin will be direction-dependent, and will therefore not exist.
The iterated limit limx→0limy→0f also fails to exist: the inner limit fails to exist because of the infinite oscillations in sin1/y.
The iterated limit limy→0limx→0f is necessarily zero because the inner limit is the limit as x→0 of x times something that is finite, namely, yx2+y2+sin1/y.
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