14.2 Videos Guide

14.2a

• Definition of a limit of a function $f$ of two variables

  • $\lim\limits_{(x,\ y)\rightarrow(a,\ b)}{f(x,\ y)=L}$ if for every $\varepsilon>0$ there is a $\delta>0$ such that if $(x,\ y)$ is in the domain of $f$ and $0<\sqrt{(x-a)^{2}+(y-b)^{2}}<\delta$ then $\left|f(x,\ y)-L\right|<\varepsilon$

Exercises:

14.2b

• Find the limit, if it exists, or show that the limit does not exist

  • $\lim\limits_{(x,\ y)\rightarrow(2,\ -1)}\frac{x^{2}y+xy^{2}}{x^{2}-y^{2}}$

  • $\lim\limits_{(x,\ y)\rightarrow(0,\ 0)}\frac{5y^{4}\cos^{2}x}{x^{4}+y^{4}}$

14.2c

• $\lim\limits_{(x,\ y)\rightarrow(0,\ 0)}\frac{xy^{4}}{x^{4}+y^{4}}$

• Reminder of the Squeeze Theorem in $\mathbb{R}^{2}$ (which also holds in higher dimensions)

  • If $f(x)\leq g(x)\leq h(x)$ on an open interval containing $a$ (except possibly at $a$) and if $\lim\limits_{x\rightarrow a}{f(x)}=\lim\limits_{x\rightarrow a}{h(x)}=L$, then $\lim\limits_{x\rightarrow a}{g(x)}=L$

Exercise:

• Determine the set of points at which the function is continuous.
  $G(x,\ y)=\ln(1+x-y)$

• Use polar coordinates to find the limit.
  $\lim_{(x,\ y)\rightarrow(0,\ 0)}{\left(x^{2}+y^{2}\right)\ln\left(x^{2}+y^{2}\right)}$