One power infinity

Calculus Level 4

$\LARGE \lim_{x \to 3} \left ( \frac {6-x}{3} \right )^{\tan \left ( \frac {\pi x}{6} \right )} = \ ?$

Give your answer to 2 decimal places.

The answer is 1.890081165.

3 solutions

Kartik Sharma
Apr 26, 2015

$\displaystyle L = \lim_{x \to 3} {(\frac{6-x}{3})}^{tan(\frac{x\pi}{6})}$

$\displaystyle ln(L) = \lim_{x \to 3} (ln(\frac{6-x}{3}))(tan(\frac{x\pi}{6})$

$\displaystyle = \lim_{x \to 3} (ln(\frac{6-x}{3}))(\frac{sin(\frac{x\pi}{3})}{cos(\frac{x\pi}{6})})$

$\displaystyle = \lim_{x \to 3} \frac{(ln(\frac{6-x}{3}))(sin(\frac{x\pi}{3}))}{cos(\frac{x\pi}{6})}$

Now we can use L'Hopital's rule and get -

$\displaystyle ln(L) = \frac{2}{\pi}$

$\displaystyle L = {e}^{\frac{2}{pi}}$

Moderator note:

Right approach. For clarity, you need to explain why you take it's natural log in the first place.

Are you telling me that I and Wolfram Alpha are wrong!? Shudders

Let me show you here

Mikal Olsen - 6 years, 1 month ago

remove the bracket in the term {pi(x)} . if u want to produce fortunate results feed (pi*x) . I guess wolfram is interpreting pi(x) as a function . wolfram is also not required over here .the problem is relatively quite easy.

Soumya Dubey - 6 years, 1 month ago

lim x->3 ((6-x)/3)^(tan(pi*x/6)) is the correct input. Yours is wrong

Raymond Lei - 6 years ago

in question it is tan(pi/6) and u have taken this here pi/3 which does not lead to 1 to power infinite form u have done it wrong the answer given is also wrong it comes out to be 0.529 that is e to power (-2/pi) check once again ..............

RAJ RAJPUT - 6 years, 1 month ago

dude if its tan(xpi/6) then its 1^infinty form so we can rewrite the Q as e^{((3-x)/3) tan(xpi/6)}=Lt x->3 e^[ (3-x)/3 * {tan(xpi/6)/(xpi/6)} xpi/6 ]=1. pls go thru it and tell me whether im correct or wrong.

Krishanu Kumar - 6 years, 1 month ago

when it becomes 1 to power infinity form then we have to approach through log but generally e to power g(x) multiply ( fx -1 ) works here also but you are wrong as you have not taken g(x) after applying the form tan(xpi/6) will also come and make 0 x infinite another inderminate form .....

and solution provided is also wrong best method for this what i prefer is to take x-3=y limit changes to y tending zero

RAJ RAJPUT - 6 years, 1 month ago

@Raj Rajput – thanks i saw what i wrote.....Ltx->3, g(x)= tan(xpi/6)= tan(pi/2) not tan(0). i applied Ltx->0 tanx/x wala formula! i get it thanks!!

Krishanu Kumar - 6 years, 1 month ago

@Krishanu Kumar – most welcome but do you don't think that above given solution by kartik sharma is wrong

RAJ RAJPUT - 6 years, 1 month ago

@Raj Rajput – in what sense?? in taking the constant of x inside tan or applying l'hospital rule?

Krishanu Kumar - 6 years, 1 month ago

@Krishanu Kumar – he has taken tan(pi/3) instead of given tan(pi/6) in qyestion

RAJ RAJPUT - 6 years, 1 month ago

@Raj Rajput – yes he must have written pi/6 instead of pi/3 in question!{minor mistake }

Krishanu Kumar - 6 years, 1 month ago

@Krishanu Kumar – but answer also comes in case of tan(pi/6) but it comes out to be e to power -2/pi and answer change its a major mistake i think

RAJ RAJPUT - 6 years, 1 month ago

hey your solution is WRONG ! it's pix/6 not 3 edit it or delete it !

A Former Brilliant Member - 4 years, 6 months ago

Puneet Mehra
Apr 28, 2015

Short trick is exp(f(x)-1)(g(x))...where..f tends to1and g tends to infinite...

Moderator note:

Can you elaborate on it?

Jun Arro Estrella
Apr 27, 2015

I will show you everyone a technique that can magically find the limit of any function. YES any function. In this problem, we see that if we plug in x=3 to the expression, we will not obtain our desired limit value.. However, If we substitute a value that is very close to 3 (but is not 3) we can obtain the result we desire. we can set up x=3.000001 or x=2.99999 . But beware if its a trigonometric function, you need to set your calculator to radian mode. Thus, done .That easy.

Moderator note:

How would you solve $\displaystyle \lim_{x \to 0} \frac 1 x$ then?