I am working on one approach without using L'Hopital.
(This is not for the original poster. Do not confuse yourself. This is just for fun for other posters).
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Directly by definition.
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We want to show,
Ignore the minus sign for now. Now try to show that .
Let be small enough so that we are working on the interval .
Now, on because on . Thus, .
This means,
.
But,
.
No L'Hopital!