Question

Use L'Hospitals Rule
lim theta rightarrow 0 \frac{ theta }{ tan(theta) }

Answer 1

I know that we are suppose to substitute the zero but with the trig functions, I am not sure what to do

Answer 2

i got 0/tan 0

Answer 3

= zero and tan 0 is just zero right?

Answer 4

l'hopital rule states that both denominator and numerator can be derivated when looking for lim if lim of both original functions is either 0 or infinite

limes of both theta and tan(theta) when theta approaches 0 are 0

now we derivate both theta and tan(theta)

\[\lim_{\theta \rightarrow 0} \frac{ \theta \prime }{ \tan(\theta)\prime } = \frac{ 1 }{ \frac{ 1 }{ \sin ^{2} \theta } }\]

can you find the limes now?

Answer 5

1/sin^2theta = sin^2theta

Answer 6

do i take the reciprocal of sin^2 theta

Answer 7

sorry, my bad. i derivated tangent wrong.
tan(theta) derivated is 1/cos^2 theta

Answer 8

ok- I dont know why i thought it was sin and cos for tan

Answer 9

\[\lim_{\theta \rightarrow 0} \frac{ \theta \prime }{ \tan(\theta)\prime } = \frac{ 1 }{ \frac{ 1 }{ \cos ^{2} \theta } }\]

Answer 10

ok

Answer 11

and then you have:
\[=\frac{ \cos ^{2}\theta }{ 1 }\]

Answer 12

do i take the recipricol of that?

Answer 13

nope, you just plug in the value of theta as is defined in the limes and see what you get

Answer 14

so cos^2 (0) = 1 right?

Answer 15

L'Hopital rule is used when you have limes of a fraction (or something that can be rewritten as fraction) but both numerator and denominator approach either 0 or infinite

Answer 16

yes!

Answer 17

ok thank you so much for your help

Answer 18

np;)
here is some extra reading if you need:
http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule

Answer 19

awsome, thanks

Answer 20

You can also use the fact that \(\tan\theta\approx\theta\) for \(\theta\) near 0, giving you
\[\lim_{\theta\to0}\frac{\theta}{\theta}\]
Which you can evaluate using l'Hopital's rule, or simplify immediately.