How can I prove $\lim_{x\to \infty} f(x) = L$ ? I tried to prove by L'Hospital's Rule, but I only proved when the limit exists... How can I prove when I am not provided that the limit exists?
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Use one-sided techniques first – Jun 08 '14 at 05:11
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L'hopital's rule works here. Notice that the condition given is equivalent to $$\lim_{x \rightarrow \infty} {(e^x f(x))' \over (e^x)'} = L$$ So an application of L'hopital gives the result.
If you wonder how this might occur to someone, if you've done first order linear differential equations it is natural when seeing $f'(x) + f(x)$ to multiply by $e^x$ and writing the result as $(e^x f(x))'$.
Zarrax
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