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Squeeze theorem (also called pinch theorem or sandwich theorem) is a theorem in calculus that states that if

g(x)\le f(x)\le h(x)

and

\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L

then

\lim_{x\to a}f(x)=L

This can be used to solve limits that would otherwise be difficult or impossible. For example, the limit

\lim_{x\to0}x^2\sin\left(\tfrac{1}{x}\right)

can be found because we know

-1\le\sin\left(\tfrac{1}{x}\right)\le1

Therefore,

-x^2\le x^2\sin\left(\tfrac{1}{x}\right)\le x^2

Since \lim_{x\to0}-x^2=\lim_{x\to0}x^2=0 , by the squeeze theorem, \lim_{x\to0}x^2\sin\left(\tfrac{1}{x}\right) must also be 0.

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