Chapter 3: Functions of Several Variables

Section 3.2: Limits and Continuity

Example 3.2.23

$$ Show that the bivariate limit at the origin for $f\=\{\begin{array}{cc}\frac{\left|x\right|}{y^2}{e}^{-\|x\|\/y^2}& y\ne 0\\ 0& y\=0\end{array}$ does not exist. $$ Solution $$ Along the parabola $x\=y^2$, $f\left(y^2\,y\right)\=\frac{y^2}{y^2}{e}^{-y^2\/y^2}\=e^{-1}$, but along the line $x\=0$, $f\left(0\,y\right)\=0$. Hence, the bivariate limit at the origin cannot exist. $$

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