I need major help with limits of trig functions

[dl146379]

I have two problems that are completely confusing me. I am very bad with trig identities.

Problem 1:
lim sin 7θ /θ + tan 8θ
θ→0

Do I change the tan to sin/cos ?

Problem 2:
lim cos 2θ − 1 / sin 3θ
θ→0

With this one, do I use one of the cos 2θ identities?

 

[tkhunny]

How do you know you are bad at these limits? Do you know that you should be learning about them and they are not intended to be trivial?

You could use a little work with "Order of Operations". Notice that A/B+C is NOT the same as A/(B+C).

You may change \(\displaystyle \tan(\theta)\) to \(\displaystyle \sin(\theta)/\cos(\theta)\) if you like. Will it help? Does it lead anywhere?

You may also wish to pursue various expansions, such as \(\displaystyle \sin(8\theta) = \sin(7\theta + \theta)\). Doea that lead anywhere?

 

[galactus]

A little trick you can use with \(\displaystyle \displaystyle\lim_{\theta\to 0}\frac{sin(7\theta)}{\theta}\)

is to multiply top and bottom by 7 and remember the useful and famous limit:

\(\displaystyle \displaystyle\lim_{\theta\to 0}\frac{sin(\theta)}{\theta}=1\)