Editing Squeeze theorem (section)

=== Fourth example ===

The squeeze theorem can still be used in multivariable calculus but the lower (and upper functions) must be below (and above) the target function not just along a path but around the entire neighborhood of the point of interest and it only works if the function really does have a limit there. It can, therefore, be used to prove that a function has a limit at a point, but it can never be used to prove that a function does not have a limit at a point.<ref>{{cite book|chapter=Chapter 15.2 Limits and Continuity| pages=909–910|title=Multivariable Calculus|year=2008|last1=Stewart|first1=James| author-link1=James Stewart (mathematician)| edition=6th|isbn=978-0495011637}}</ref>

<math display="block">\lim_{(x,y) \to (0, 0)} \frac{x^2 y}{x^2+y^2}</math>

cannot be found by taking any number of limits along paths that pass through the point, but since

<math display="block">\begin{array}{rccccc}
  & 0 & \leq & \displaystyle \frac{x^2}{x^2+y^2} & \leq & 1 \\[4pt]
  -|y| \leq y \leq |y| \implies & -|y| & \leq & \displaystyle \frac{x^2 y}{x^2+y^2} & \leq & |y| \\[4pt]
  {
    {\displaystyle \lim_{(x,y) \to (0, 0)} -|y| = 0} \atop 
    {\displaystyle \lim_{(x,y) \to (0, 0)} \ \ \ |y| = 0}
  } \implies & 0 & \leq & \displaystyle \lim_{(x,y) \to (0, 0)} \frac{x^2 y}{x^2+y^2} & \leq & 0
  \end{array}</math> 

therefore, by the squeeze theorem,

<math display="block">\lim_{(x,y) \to (0, 0)} \frac{x^2 y}{x^2+y^2} = 0.</math>