Help with these limits!!

Nov 2019
7
0
California
a) lim x--> 0+ (sin (x))^x

b) lim x-->infinity (pi/2 - arctan(x))^x
 
Dec 2015
1,086
170
Earth
\(\displaystyle a^b =e^{b\ln(a)}\).
 
Nov 2019
7
0
California
ok I see how the expression you gave can solve the first problem. However, if I do it for the second one, I'm still stuck with ln(pi/2-arctan(x)) and this would give me ln(0) which is -infinity???
 
Dec 2015
1,086
170
Earth
\(\displaystyle \lim_{x\rightarrow \infty} (\pi /2 - \arctan(x))^{x} =\lim_{x\rightarrow \infty} (1+[\pi /2 - \arctan(x)-1])^{\displaystyle x\cdot \displaystyle \frac{\pi/2 - \arctan(x)-1}{\pi/2 - \arctan(x)-1}}=\lim_{x\rightarrow \infty} e^{x(\pi /2 -\arctan(x)-1)}=\lim_{x\rightarrow \infty}e^{-x}=0\).
 
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skipjack

Forum Staff
Dec 2006
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What is \(\displaystyle \lim_{x\to \infty} (\pi /2 - \arctan(x) + 1)^{x}\)?
 

greg1313

Forum Staff
Oct 2008
8,008
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London, Ontario, Canada - The Forest City
\(\displaystyle t=\frac\pi2-\arctan x\)

\(\displaystyle x=\tan\left(\frac\pi2-t\right)\)

\(\displaystyle \lim_{t\to0}t^{\tan(\pi/2-t)}=0^1=0\)
 
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Dec 2015
1,086
170
Earth
(b) \(\displaystyle 0^\infty =0\).