l'hopital Question

Jan 2010
22
0
Hiii :3

I'm stuck on this question:


lim (x->1) (2-x)^ tan(π/2)x


I did the ln thing but now i'm stuck what do I do next?
Plug it in?



Thank youuus :]
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
Hiii :3

I'm stuck on this question:


lim (x->1) (2-x)^ tan(π/2)x


I did the ln thing but now i'm stuck what do I do next?
Plug it in?



Thank youuus :]
Seeing as you've been told to use L'Hospital's Rule, it would make sense to use L'Hospital's Rule, wouldn't it?


You are correct that you should apply the exponential-logarithmic transformation...

\(\displaystyle \lim_{x \to 1} (2 - x)^{\tan{\frac{\pi x}{2}}} = \lim_{x \to 1}e^{\ln{(2 - x)^{\tan{\frac{\pi x}{2}}}}}\)

\(\displaystyle = \lim_{x \to 1}e^{\tan{\left(\frac{\pi x}{2}\right)}\ln{(2 - x)}}\)

\(\displaystyle = \lim_{x \to 1}e^{\frac{\sin{\left(\frac{\pi x}{2}\right)}\ln{(2 - x)}}{\cos{\left(\frac{\pi x}{2}\right)}}}\)

\(\displaystyle = e^{\lim_{x \to 1}\frac{\sin{\left(\frac{\pi x}{2}\right)}\ln{(2 - x)}}{\cos{\left(\frac{\pi x}{2}\right)}}}\).


This limit is now of the form \(\displaystyle \frac{0}{0}\), so now you can use L'Hospital's Rule.
 
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