Question

lim x->0 [x*cot((pi*x)/3)]. Why is the answer 3/pi ?

Answer 1

do u know lhopitals rule?

Answer 2

\[\frac{x\cot(\pi x)}{3}=\frac{x\cos(\pi x)}{\sin(\pi x)}\]
\[=\frac{\pi x}{\pi \sin(\pi x)}\times \cos(\pi x)\] will work if you cannot use l'hopital

Answer 3

lol i love lhopitals ... Nice @satellite73

Answer 4

how bout them zeta functions?

Answer 5

i misses a 3 somewhere in my last expression
\[=\frac{\pi x}{3\pi \sin(\pi x)}\times \cos(\pi x)\]
now let \(x\to 0\) and you should get \(\frac{\pi}{3}\) because \(\cos(0)=1\) and \(\lim_{x\to 0}\frac{\sin(x)}{x}=1\)

Answer 6

damn i am making mistake after mistake

Answer 7

i think the answer is \(\frac{1}{3\pi}\)

Answer 8

its x*cot((pi*x)/3) not what you said, which was x*cot((pi*x))/3

Answer 9

lets start here
\[\frac{\pi x}{3\pi \sin(\pi x)}\times \cos(\pi x)\]

Answer 10

\[\lim_{x\to 0}x\cot(\frac{\pi x}{3})\]

Answer 11

your last one is correct

Answer 12

ok same idea as last one

Answer 13

\[x\cos(\frac{\pi x}{3})=\frac{x\cos(\frac{\pi x}{3})}{\sin(\frac{\pi x}{3})}\]

Answer 14

multiply top and bottom by \(\frac{\pi}{3}\)

Answer 15

\[\frac{\frac{\pi}{3}x\cos(\frac{\pi x}{3})}{\frac{\pi}{3}\sin(\frac{\pi x}{3})}\]

Answer 16

then \[\lim_{x\to 0}\cos(\frac{\pi x}{3})=\cos(0)=1\]

Answer 17

and \[\lim_{x\to 0}\frac{\frac{\pi}{3}x}{\sin(\frac{\pi x}{3})}=1\]

Answer 18

leaving you with
\[\frac{1}{\frac{\pi}{3}}=\frac{3}{\pi}\]

Answer 19

why is this true?: xcos(πx/3)=xcos(πx/3)sin(πx/3)

Answer 20

it isn't. you had cotangent, not cosine right?