Editing Squeeze theorem (section)

=== First example ===

[[File:Inst_satsen.svg|thumb|right|250px|<math>x^2 \sin\left(\tfrac{1}{x}\right)</math> being squeezed in the limit as {{mvar|x}} goes to 0]]

The limit

<math display="block">\lim_{x \to 0}x^2 \sin\left( \tfrac{1}{x} \right)</math>

cannot be determined through the limit law

<math display="block">\lim_{x \to a}(f(x) \cdot g(x)) = 
\lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x),</math>

because

<math display="block">\lim_{x\to 0}\sin\left( \tfrac{1}{x} \right)</math>

does not exist.

However, by the definition of the [[sine function]],

<math display="block">-1 \le \sin\left( \tfrac{1}{x} \right) \le 1. </math>

It follows that

<math display="block">-x^2 \le x^2 \sin\left( \tfrac{1}{x} \right) \le x^2 </math>

Since <math>\lim_{x\to 0}-x^2 = \lim_{x\to 0}x^2 = 0</math>, by the squeeze theorem, <math>\lim_{x\to 0} x^2 \sin\left(\tfrac{1}{x}\right)</math> must also be 0.