The Squeeze Theorem

The basic idea behind this theorem is quite intuitive, and it would be quite surprising if this theorem wasn’t true:

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Suppose that  and , and further suppose that  for all x in some open interval that contains L.   Then

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We'll  illustrate this with one rather obvious example, followed by an example less obvious but much more important.

Example 1  

Let  f(x) = x² sin x.   Since the sine function always has absolute value less than or equal to 1, it follows that:   

– x² <  x² sin x <  x² .    

Since  it follows that      

 

Example 2      

 

Consider the following graph of part of the unit circle in Quadrant 1:

The angle here is in radians (as usual!).
The yellow region has area (half base times height)  (1/2)(cos x)(sin x).
The yellow and turquoise regions all together have area x/2.

The yellow, turquoise, and red regions all together have area (1/2)(1)(tan x).

Thus we must have

                                             

Multiplying by 2/sin x, this becomes (for small positive values of x; a similar argument works for small negative values of x):

                                                       

Now take the limit as x goes to 0:

                                            

 

 

Invoking the squeeze theorem, we see that  

Thus also    by limit property [5]  (with r = –1).