# Sandwich Theorem

## Related Calculator: Limit Calculator

Fact. If f(x)<=g(x) when x is near a (except possibly at a) then lim_(x->a)f(x)<=lim_(x->a)g(x).

This fact means that if values of f(x) are not larger than values of g(x) near a, then f(x) approaches not larger limit than g(x) as x->a.

Sandwich Theorem (or Squeeze Theorem). Consider three functions f(x),g(x),h(x). If we have that f(x)<=g(x)<=h(x), when x near a (except possibly at a) and lim_(x->a)f(x)=lim_(x->a)h(x)=L then lim_(x->a)g(x)=L.

This theorem tells us folowing: if there are three functions, two of which have same
limit as x approaches a and third is "squeezed" between them, then third will have to approach same limit as x approaches a as first two.

Example. Find lim_(x->0)x^2cos(1/x).

Since -1<=cos(1/x)<=1 for all x (actually we are interested only in x near 0) then -x^2<=x^2cos(1/x)<=x^2. Since lim_(x->0)x^2=lim_(x->0)-x^2=0 then by Squeeze theorem lim_(x->0)x^2cos(1/x)=0.

On the figure you can see that x^2cos(1/x) is squeezed between x^2 and -x^2.