Squeeze Theorem
The Squeeze theorem is a part of mathematics and study under Calculus. and we all know what calculus is, and why and where we use it. In Calculus, we study about the swap in between the values which are interrelated with the function. It is used for analysation of mathematics and in calculus. Students find this theorem sometimes very arduous, But not to worry we are here to help you out in understanding the Squeeze theorem and tells you the uncomplicated ways to solve the problems of this theorem. Which not only clear your thoughts on Squeeze theorem but also saves your time and also gives you the accurate results of your findings.
What is Squeeze Theorem
Basically in Squeeze theorem, is used to find out the functions limit or the limit of the function. When we know the limits of the more functions and when these more functions can be calculated easily we compare it with a limit of function or functions limit to confirm it. Types of Squeeze theorem OR Various names of this theorem. it sometimes sounds funny but it is a fact that squeeze theorem has various different kinds of the name like.
- Sandwich Theorem
- Policeman Theorem
- Pinching Theorem
- Theorem of Carabinieri
Understanding Squeeze Theorem
In Squeeze Theorem, we always assume three functions which are like as follow f(x), g(x), h(x). And now how these functions are written it is written in this format diagram 1f of x, f(x) is less than or equal to g of x, g(x), is less than or equal to h of x, h(x) this how we tell the relation between these three functions.
Representation in graphical format
when we represent these three functions in the graph we saw this kind of a graph which is showing us that, we draw a graph x-axis and y-axis with origin as O, now from y-axis we have curved lines which are representing m is equal to f of x, m = f(x), m is equal to g of x, m = g(x), and m is equal to h of x, m = h(x). now if we draw a point and it is x is equal to a, x = a somewhere on the x-axis as here we draw a point in the centre of the x-axis. What we are analyzing we analysed that m is equal to f of x, m = f(x), and m is equal to g of x, m = g(x), and m is equal to h of x, m = h(x) is meeting on the same point ‘a’ and these functions are very close and also the difference is very little as it is seen in the diagram. In the diagram, it is identified that when functions m= f(x), m = g(x), m = h(x) are moving towards x is equal to a, x = a the difference is very less and close and lying on the same point, the diagrammatical Squeeze Theorem representation somehow clears the point which we are trying to understand.
Applying limits to the functions
here lim denotes the limit of the functions
lim x → a f (x) = l
We say limit x goes to a, f of x is equal to l
lim x → a h (x) = l
We say limit x goes to a, h of x is equal to l
lim x → a g (x) = l
We say limit x goes to a, g of x is equal to l now, lim x → a f (x) = l, we have some value and that is l,
lim x → a, h (x) = l, here also we have the value l and the value of x → a f (x) = l, x → a h (x) = l, is identical or in simple language, we say it is that the value in both the functions is the same whether it is f of x or h of x the value is same, now we can say that limit x goes to a g of x is also equal to l, x → a g (x) = l. This is only because the x → a g (x) = l, lies in between the limit x → a h (x) = l and limit x → a f (x) = l and as we say and seen on the graph, that all the functions m = f(x), m = g(x), m = h(x) is lying on the same point and are very closer. m = f(x), m = g(x), m = h(x) , the limit m = g (x) is lying in between that’s why it is said that it has same value as the value of m = f(x) and m = h(x). This the way applied the limit to the functions of Squeeze Theorem that you may love to learn about.