SOLUTION: Evaluate the limit (as x approaches to 0+) x^4sin(1/x)

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Question 1200072: Evaluate the limit (as x approaches to 0+) x^4sin(1/x)
Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.

The given function is limited from the top by the function  y%5B1%5D = x%5E4

and is limited from the bottom by the function  y%5B1%5D = -x%5E4


    -x%5E4 <= x%5E4%2A%28sin%281%2Fx%29%29 <= x%5E4.


Both functions  y%5B1%5D  and  y%5B2%5D  have the limit 0 (zero) as x approaches to 0+.


Therefore, the given function  x%5E4%2A%28sin%281%2Fx%29%29  has the limit 0 (zero)  as x approaches to 0+.

Solved, with complete explanations.


The referenced theorem of Calculus has a joking folklore name "two policemen theorem".

It says "if two police officers are bringing an intoxicated prisoner between them to a cell, the prisoner must also end up in the cell".

In our problem, two limiting functions do represent two police officers,
while the given function does represent an intoxicated prisoner.


See this link
https://en.wikipedia.org/wiki/Squeeze_theorem

together with the accompanying figures.


One of these figures is close visual presentation to your function and my limiters.