Evaluate the following limits, assuming all angles are in radian

a) lim x->0 (sin5x)/(sin6x)

b) lim x->0 (xsin6x)/(sin^2(9x))

c) lim x->0 (sin3x)/(9x-2tanx)

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- Oct 13th 2009, 05:26 PMkatekateLimits, radian angles
Evaluate the following limits, assuming all angles are in radian

a) lim x->0 (sin5x)/(sin6x)

b) lim x->0 (xsin6x)/(sin^2(9x))

c) lim x->0 (sin3x)/(9x-2tanx) - Oct 13th 2009, 05:32 PMartvandalay11
do you know l'hospital's rule?

- Oct 13th 2009, 05:36 PMkatekate
yeah i know l'hopital's rule...i definitely did not understand the second part you said but if i use l'hopital it should work

- Oct 13th 2009, 05:40 PMartvandalay11

lol that's just my signature and has nothing to do with you or this post, but let me do the first one and have you try the second one

$\displaystyle \lim_{x\rightarrow 0}\frac{\sin(5x)}{\sin(6x)}$ is indeterminant, meaning you get $\displaystyle \frac{0}{0}$ if you try to plug into the expression, so we apply L'hospital's rule to get

$\displaystyle \lim_{x\rightarrow 0}\frac{\sin(5x)}{\sin(6x)}=\lim_{x\rightarrow 0}\frac{5\cos(5x)}{6\cos(6x)}=\frac{5\cos(0)}{6\co s(0)}=\frac{5}{6}$

do the same thing for the next - Oct 13th 2009, 05:55 PMkatekate
I can't get the second one but I get 3/5 for the third. I think im messing up the derivatives of both top and bottom.

**Edit:**never mind 3/5 is definitely wrong too - Oct 13th 2009, 06:15 PMartvandalay11

alright so let's do this

LH means i used l'hospital's rule

$\displaystyle \lim_{x\rightarrow 0}\frac{x\sin(6x)}{\sin^2(9x)}=LH=\lim_{x\rightarr ow 0}\frac{\sin(6x)+6x\cos(6x)}{2\cdot 9\sin(9x)\cos(9x)}$

Since we still get $\displaystyle \frac{0}{0}$ we use LH again

$\displaystyle =\lim_{x\rightarrow 0}\frac{6\cos(6x)+6\cos(6x)-36x\sin(6x)}{162\cos^2(9x)-162\sin^2(9x)}=\frac{12\cos(0)}{162\cos^2(0)}=\fra c{12}{162}=\frac{2}{27}$

any questions?

try the third one more time before I help you out there too - Oct 13th 2009, 06:23 PMkatekate
great i got both of them now, i was squaring the 2 in the denominator when i shouldn't have been before in the third one. i didnt realize i had to use l'hopitals more than once, thanks a ton for everything