Question

What is the mistake in these steps ?

when I'm using Squeeze theorem it doesn't work. why ?

`Limx-> oo cos(1/x)/(1+1/x)`

`-1<cos(1/x)<1`

`-1/(1+1/x)<cos(1/x)/(1+1/x)<1/(1+1/x)`

-1 1

so It will be -1 and 1 so the limit doesn't exist ?

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Answer 1

Please see below.

Explanation:

First, we need "or equal to" `-1 <= cos(1/x) <=1`

`-1/(1+1/x) <= cos(1/x)/(1+1/x) <= 1/(1+1/x)` is OK

Now taking limit as `xrarroo`, we get

`-1 <= lim_(xrarroo)cos(1/x)/(1+1/x) <= 1`

Which is true, but not helpful to find the limit. We know that if it exists, it is between `-1` and `1`. But that does not show that the limit exists and it does not tells us what the limit is.