Question

I am learning calculus, and have been stuck on this question. I cannot use Ls Hospital Rule to solve it and i’m stuck after one step

Answer 1

Using the trigonometric identities given, we get:

$\begin{gathered} \lim _{\theta\rightarrow0}\text{ }\frac{\cos(\theta+A)-\cos(\theta-B)}{\theta}=\lim _{\theta\rightarrow0}\frac{\cos \theta\cos A-\sin \theta\sin A-(\cos \theta\cos B-\sin \theta\sin B)}{\theta} \\ =\lim _{\theta\rightarrow0}\frac{\cos \theta(\cos A-\cos B)+\sin \theta(\sin B-\sin A)}{\theta} \\ =\lim _{\theta\rightarrow0}\frac{\cos\theta}{\theta}(\cos A-\cos B)+\lim _{\theta\rightarrow0}\frac{\sin \theta}{\theta}(\sin B-\sin A) \end{gathered}$

Recall that:

$\lim _{\theta\rightarrow0}\frac{\cos\theta}{\theta}\text{ does not exist.}$

And the limit of the product is the product of the limits.

Therefore the limit of interest does not exist.