# Limit - which result is correct?

by twoflower
Tags: correct, limit, result
 P: 368 Hi all, I've been practising for the exam and did some limits from our master's collections: $$\lim_{x \rightarrow \frac{\pi}{3}} \frac{\tan^{3} x - 3\tan x}{\cos \left( x + \frac{\pi}{6} \right) }$$ I got the result -24, but there is written -12. Which one is correct? I tried to prove the result in Maple and Mathematica, but in neither I'm not able to get anything but symbolic result. Thank you.
 Sci Advisor HW Helper P: 11,896 HINT:Bring your function to anither form,so that the limit can be put under the "conventional" form "->0".That requires a change of variable. Daniel. P.S.Can as in do you know how to apply L'Hôspital's rule??
 P: 368 Ok, so now I put the "L'Hospitaled" function to the Maple and it still gives me symbolic result...
 Sci Advisor HW Helper P: 11,896 Limit - which result is correct? Can Maple compute limits??Can u compute derivatives??CAN U BOTH SOLVE THIS SIMPLE PROBLEM????? Daniel.
HW Helper
Thanks
P: 10,383
 Quote by twoflower I got the result -24, but there is written -12. Which one is correct?
Your result is correct.

Just try to choose some value for x close enough to pi/3 and calculate the replacement value. You will see that it is far away from -12.

ehild
P: 368
 Quote by dextercioby Can Maple compute limits??Can u compute derivatives??CAN U BOTH SOLVE THIS SIMPLE PROBLEM????? Daniel.
I don't understand. Using L'Hospital I got the result I wrote here. But the "official" result is different to mine. I was just asking how to prove it in Maple.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,352 The point was: Why use MAPLE?? You can't PROVE the MAPLE calculation is correct by using MAPLE! It's easy to calculate that the derivative of tan3(x)- 3tan(x) is 3tan(x)sec2(x)- 3sec2(x) and certainly you should know that $tan(\frac{\pi}{3})= \sqrt{3}$ and that $sec(\frac{\pi}{3}})= 2$. Of course, the derivative of $cos(x+\frac{\pi}{6}$ is $-sin(x+\frac{\pi}{6})$ and $-sin(\frac{\pi}{2}})= -1$. The limit, by L'Hopital, is (3*3*4- 3(22)/-1= -(36-12)= -24.
P: 368
 Quote by HallsofIvy The point was: Why use MAPLE?? You can't PROVE the MAPLE calculation is correct by using MAPLE!
I didn't want to prove the MAPLE calculation, I was trying to prove the teacher's calculation...
HW Helper
P: 2,887
 Quote by HallsofIvy The point was: Why use MAPLE?? You can't PROVE the MAPLE calculation is correct by using MAPLE! It's easy to calculate that the derivative of tan3(x)- 3tan(x) is 3tan(x)sec2(x)- 3sec2(x) and certainly you should know that $tan(\frac{\pi}{3})= \sqrt{3}$ and that $sec(\frac{\pi}{3}})= 2$. Of course, the derivative of $cos(x+\frac{\pi}{6}$ is $-sin(x+\frac{\pi}{6})$ and $-sin(\frac{\pi}{2}})= -1$. The limit, by L'Hopital, is (3*3*4- 3(22)/-1= -(36-12)= -24.
That's indeed correct (notice that there's a typo, it should obviously be

"It's easy to calculate that the derivative of
tan3(x)- 3tan(x) is 3tan2(x)sec2(x)- 3sec2(x)"

i.e. there is a square in the first tan in the derivative but it's obviously a typo because HallsofIvy gave the correct numerical answer.

Just a thought: if someone forgets to square the sec(x), they will get -12. So that could be the origin of the incorrect answer -12.

Pat
 HW Helper Thanks P: 10,383 The limit can be obtained even without L'Hopital's rule. $$\lim_{x \rightarrow \frac{\pi}{3}} \frac{\tan^{3} x - 3\tan x}{\cos \left( x + \frac{\pi}{6} \right) }= \lim_{x \rightarrow \frac{\pi}{3}} \frac{\tan(x)(\tan^{2} x - 3)}{\cos ( x )\cos (\pi/6)-\sin(x)\sin(\pi/6)}=$$ $$\lim_{x \rightarrow \frac{\pi}{3}}\frac{\tan(x)(tan(x)-\sqrt(3))(\tan(x)+\sqrt(3))}{1/2\cos(x)(\sqrt(3)-\tan(x))}= \lim_{x \rightarrow \frac{\pi}{3}}\frac{-2\tan(x)(\tan(x)+\sqrt(3))}{\cos(x)}=\frac{-2\sqrt(3)(2\sqrt(3)}{1/2}=-24$$ ehild
P: 368
 Quote by ehild The limit can be obtained even without L'Hopital's rule. ehild
Nice ehild, I thought this is one of the limits which can't be solved without L'Hospital's rule...Thank you.

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