Question

Evaluate the following limit or explain why they do not exist.

lim as theta approaces pi/2^- tantheta^cos(thetha)

Answer 1

I'm not sure if i'm doing this correctly, the problem is:
\[\lim_{\theta \rightarrow \pi/2^-} (\tan \Theta)^{\cos \Theta}\]
USing this formula
\[b^{x}= e^{xlnb} \]
I get
\[\cos \Theta \ln(\tan \Theta)\]
According to my steps for L Hopital:
1. Find indeterminate form
2. Get ratio
3. get derivative, no quotient rule used for the ratio
4. Find the limit
5. repeat if in indeterminate form

Ratio:
\[\ln(\tan \Theta)/\sec \Theta = f \prime/ g \prime = \frac {\sec^2 \Theta/\tan \Theta}{\sec \Theta \tan \Theta}\]
Is this right so far?

Answer 2

*that should be ratio and the derivative

Answer 3

I also had a question about why is it that when I evaluate:
\[\sec^2 \Theta \] as Theta approaches pi/2^- the answer is infinity, I thought it would be 0 since costheta at pi/2^- is zero.

Answer 4

sec and cos are reciprocals of one another

Answer 5

cos =0 then sec=1/0 (or does not exist)

Answer 6

and yes you are right so far
do another thing of l'hospital

Answer 7

before you do it make sure you clean up that nasty fraction.