Question

Calculus question!
lim f(θ) where f(θ) = {3cos(2θ ) if θ ≤ π
θ→ π+ { (sin(θ/2)) / (θ) if θ>π


Can anyone show me a step by step solution how to do this so I know how to do the other ones similar to it? Please and thank you!

All I know is that I am supposed to use the (sin(θ/2)) / (θ) if θ>π equation. So what do I do next?

Answer 1

You want to do the left hand and right hand limits.

Answer 2

How would I do that?

Answer 3

\[
\lim_{\theta\to \pi^+}3\cos(2\theta) = 3\cos(2\pi) = 3
\]

Answer 4

\[
\lim_{\theta\to \pi^-}\frac{\sin(\theta/2)}{\theta} = \frac{\sin(\pi/2)}{\pi} = \frac 1 \pi
\]

Answer 5

\[
\lim_{\theta\to\pi^-}f(\theta) \neq \lim_{\theta\to\pi^+}f(\theta)
\]The limit does not exist.

Answer 6

Oh! I see, now it makes sense! Thank you so much! SO because the left and right hand sides are not equal to each other, the limit does not exist?

Answer 7

That is a basic fact about limits.