Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain wh

Question

Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s
Rule doesn’t apply, explain why.

 

lim (theta --> pi/2)   (1 - sin(theta))/(1 + cos(2theta))

 

ANSWER: 1/4

p307, #15

Answer 1

Hi Mia.

 

I am going to use x in this discussion (rather than theta).

 

As x approaches pi/2 , the function approaches 0/0 which in turn means that L'Hospitals rule can be applied. So taking the derivative of the numerator over the derivative of the denominator generates an expression that will approach the same limit as the original. Normally I would show the Lim as x-> pi/2 for each step of the following work, but will leave it out here.

 

Here is what we get after using the rule:

 

- cos (x) / - sin (2x)(2) or cos x/ 2sin (2x). At this point it might be an OK idea to see if this can be simplified using the double angle formula for sin (2x) = 2sinxcosx.

 

cosx/(2)(2sinxcosx) reduces to 1/4sinx.

 

lim_x->pi/2(1/4sinx) = 1/4(1) = 1/4 (by direct substitution of pi/2)