Pinching Theorem

 

Pinching Theorem is very useful while solving some complicated problems for finding limits of some functions. But by using the Pinching theorem we can be easily solved. Pinching Theorem is also called as Sandwich theorem or it can also be called as Squeeze theorem. The names are different but all refers to the same theorem. Here we are going to see about pinching theorem.

 

 

Pinching Theorem:

If g (x) ≤ f (x) ≤ h (x) and this is true for all x at  x = a.

If, suppose

 `lim_(x->a)` g (x) =  `lim_(x->a)`h ( x ) = L.

Then, `lim_(x->a)` = f ( x )

                   f (x) = L.

Here f (x) is in between g (x) and h (x) therefore this is said to be a sandwich theorem

Because the limits are sandwiched between two values so it is called as sandwich theorem.

Example Problem - 1 for Pinching Theorem:

 

Prove that`lim_(a->0)` ` [ 1 - cos a ] / ( a )`   = 0  by using pinching theorem.
       

Solution for pinching theorem:

Given Problem to prove


`lim_(a->0)`    ` [ 1 - cos a ] / ( a )`    = 0
    

 `lim_(a->0)`      ` [ 1 - cos a ] / ( a )`   

`lim_(a->0)`     =   ` (1 - cos a) / ( a )`  ` (1 + cos a ) / (1 + cos a)`  (Here we are rationalizing the given expression)
                                                                                     
                
`lim_(a->0)`       `(1 - cos2(a)) /( a(1 + cos a)) `

`lim_(a->0)`   ` (sin2(a) ) / ( a(1 + cos a) )`  (Here we use a trigonometric identity)

                                            
`lim_(a->0)`     `(sin a . sin a) /( a(1 + cos a)) `  (Here we separate the fractions sin2x as sin x and sin x)
               

                                                       
`lim_(a->0)`     `(( sin a ) ( sin a) )/ (a (1 + cos a))`    (Here we are separating the limits to each fraction)

we get it as`lim_(a->0)`     `sin a / a`                   

`lim_(a->0)`   `sin a / (1 + cos a) `       (limits of product  = product of limits)
                                                                          
1 `lim_(a->0)`     `sin a / (1 + cos a)`       (Here we are using the limit value as function)

 1 `lim_(a->0)`  ` sin 0 / (1 + cos 0)`  
       
`*` `lim_(a->0)`  `0 / (1 + 1)`

             
1 * 0 = 0 

The given problem is proved using the pinching theorem:

Example Problem - 2 for Pinching Theorem:

Prove that `lim_(a->0)` a cos (1/a) = 0. by using pinching theorem.

 

 

Solution for pinching theorem:

For any value of a and a ≠ 0 we have

 `|`a cos (1/a)`|``|`  

Hence  there is an absolute value we get it as follows,

- `|` a `|` ≤ a cos(1 / a)   ≤ + `|` a `|`

`lim_(a->o)``|` a `|`    =  `lim_(a->o)``|` a `|`    =  0

then the Sandwich Theorem says that

`lim_(a->o)`    x cos (1/a ) = 0

Hence the given problem is proved by using pinching theorem.