calculus - Limits for sine and cosine functions

Recently I took a test where I was given these two limits to evaluate:
$\lim_\limits{h \to 0}\frac{\sin(x+h)-\sin{(x)}}{h}$ and $\lim_\limits{h \to 0}\frac{\cos(x+h)-\cos{(x)}}{h}.$ I used sine and cosine addition formulas and found the value of each limit individually, eventually canceling out $\sin x\cdot \frac1h$ and $\cos x\cdot \frac1h$ because I learned that we can evaluate limits piece by piece. As a result, I got $\cos x$ and $-\sin x$ as my two answers. However, my teacher marked it wrong saying that we cannot cancel $\sin{x}\cdot\frac1h$ or $\cos{x}\cdot\frac1h$ because those limits do not exist. Can someone please explain why this doesn't work? I thought that we can cancel those limits out since we never look at $0,$ just around $0,$ when evaluating these two limits.

Sine: $$\frac{\sin(x+h)-\sin(x)}h=\frac{\sin(x)\cos(h)+\sin(h)\cos(x)}h-\frac{\sin(x)}h$$

$$=\sin(x)\frac1h+\cos(x)-\sin(x)\frac1h=\cos(x)$$

Cosine: $$\frac{\cos(x+h)-\cos(x)}h=\frac{\cos(x)\cos(h)-\sin(x)\sin(h)}h-\cos(x)\frac1h$$

$$=\cos(x)\frac1h-\sin(x)\cdot1-\cos(x)\frac1h=-\sin(x)$$

Note: I am allowed to assume $\lim_\limits{x\to 0} \frac{\sin(h)}h=1,\lim_\limits{x\to 0} \frac{\cos(h)-1}h=0.$

Answer

You broke up the limits incorrectly. That can be done only when the individual limits exist. $\color{red}{\lim_\limits{h \to 0} \frac{\sin x}{h}}$ and $\color{red}{\lim_\limits{h \to 0}\frac{\cos x}{h}}$ do NOT exist.

Here is how you can solve the first limit properly.

$$\lim_{h \to 0}\frac{\sin(x+h)-\sin x}{h}$$

$$= \lim_{h \to 0}\frac{\color{blue}{\sin x\cos h}+\cos x\sin h\color{blue}{-\sin x}}{h}$$

$$= \lim_{h \to 0}\frac{\color{blue}{\sin x(\cos h-1)}+\cos x\sin h}{h}$$

$$= \lim_{h \to 0}\frac{\sin x(\cos h-1)}{h}+\lim_{h \to 0}\frac{\cos x\sin h}{h}$$

$$= \sin x\cdot\lim_{h \to 0}\frac{\cos h-1}{h}+\cos x\cdot\lim_{h \to 0}\frac{\sin h}{h}$$

Using $\lim_\limits{h \to 0} \frac{\sin h}{h} = 1$ and $\lim_\limits{h \to 0}\frac{\cos h-1}{h} = 0$, you get

$$= \sin x\cdot 0 + \cos x\cdot 1 = \cos x$$

Notice how the individual limits exist. Therefore, the procedure is correct. Can you use the same way to solve for the second limit?