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Differential Calculus
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@silane : i am a gud crammer.....and statements to as it is yaad h.....cant help it ;D......SB sir ke hi notes h....anyways.....if i am rite.....this theorem is basically used for subjective approach ???.....upto wot i remember from lectures ;)....u agree??
@ aman : this theorem is not as much used as other ones....dont panic too much about this...!!
The squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich rule and sometimes the squeeze lemma) is a theorem regarding the limit of a function.
The squeeze theorem is a technical result which is very important in proofs in calculus and mathematical analysis. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Gauss.
In Italy, Russia and France, the squeeze theorem is also known as the two carabinieri theorem, two militsioner theorem, two gendarmes theorem, or two policemen and a drunk theorem. The story is that if two police officers are holding a prisoner between them, and both the officers are going to the cell, the prisoner must also be going to the cell.
The squeeze theorem is formally stated as follows.
Let I be an interval containing the point a. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:
and also suppose that:
Then
- The functions g(x) and h(x) are said to be lower and upper bounds (respectively) of f(x).
- Here a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits.
- A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits as x → ∞.
Proof. From the above hypotheses we have, taking the limit inferior and superior:
so all the inequalities are indeed equalities and the thesis immediately follows.
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STATEMENT FOR SANDWICH THEOREM : if 'f', 'g' , 'h' are three f:ns such dat f(x) <= g(x) <= h(x) for every x in some intervel containing x = c and if [ limit f(x) {x --> c) =L ] and [ limit h(x){x-->c} = L ] then according to squeeze play theorem [ limit g(x) {x-->>c} = L]....!!
NOTE : 'c' and 'L' can be finite or infinite!!