Note:
there is a problem with the above example.
We may not know it is "legal" to evaluate the limit inside a function, unless we know for sure that the limit exists and the function is continuous.
Here we try not to prove that the function ln(x) is in fact continuous.
Instead, we make use of the inverse function of ln(x) is exp(x) and try to do the trick together with chain rule.
In the following demonstrations, let x>0.
Recall that
so ,
we can show that
By Squeeze theroem,
Finally we use the inverse function of exp(x) to do the trick :
More troubles:
1. Why exp(x) is the inverse function of ln(x) or why ln(x) an inverse function of exp(x) ?
2. What is chain rule?
3. Why I can switch the limit from h->0 to m->infinity?
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