granica w punkcie
: 27 lis 2011, 15:44
\(\lim_{x\to 1} x^{ctg(1-x)}\)
\(\lim_{x\to 1} x^{ctg(1-x)}=\lim_{x\to 1} e^{ ln \left(x^{ctg(1-x)} \right) }=\lim_{x\to 1} e^{ {ctg(1-x)} \cdot ln x }=\lim_{x\to 1} e^{ \frac{ln x }{tg(1-x)} }=(*)\)
\(\lim_{x\to 1} \frac{ln x }{tg(1-x)} =^H\lim_{x\to 1} \frac{ \frac{1}{x} }{- \frac{1}{cos^2(1-x)} }= \lim_{x\to 1} \frac{-cos^2(1-x)}{x}=-1\)
No to \((*)=e^{-1}= \frac{1}{e}\)