Limits at a point with trigonometry

Question

Find: $\lim _{m\to 0} (\cos(x+m)-\cos(x))/(m)$

Use: $$\cos(x+m)=\cos(x)\cos(m)-\sin(x)\sin(m)$$

The answer is in terms of $x$.

I know you should plug in $\cos(x)\cos(m)-\sin(x)\sin(m)$ for $\cos(x+m)$, but I do not know where to go from there in terms of manipulating the fraction.

Any help is appreciated! :)

Answer 1

Hint:

If $f$ is a differentiable at $y$ then: $$\lim_{h\to 0}\dfrac{f(y+h)-f(y)}{h}=f’(y)$$

Answer 2

Use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

$$\cos(x+h)-\cos x=-\sin\dfrac{2x+h}2\sin\dfrac h2$$

Now use $\lim_{y\to0}\dfrac{\sin y}y=1$