Prove Limit Existence: sin(1/x) as x→0

[aquitaine]

In general, what is a good method to prove whether or not a limit exists? For example limit of sin(1/x) as x approaches zero.

 

[LuisVela]

Hi, ill recommend you to take a look to the definition of limit:
For every 'e'>0,exists 'd'>0, such that for /x-x0/<d then
/f(x)-L/<e
If that holds, then the function f(x) has limit at the point x0,
and it is equal to L.
Of course a bold statement like this doesn't seem to help to much, so here is what you do for practical purpouses:
the function 1/x has a discontinuity at x=0, so evaluate the
rightsided and leftsided limits at x=0:
lim x->0-...1/x=-inf, and lim x->0+...1/x=+inf.
Then the definition of limit doesn't hold for x0=0 and therefor the limit doesn exists at this point.
Hope i helped you.
Have a good day.


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