Why can we calculate trigonometric limits by substitution? I have always wondered why something like these $\displaystyle \lim_{x\to 0} \sin x=0$ and $\displaystyle \lim_{x\to 0} \cos x=1$ are true.
Why should this happen?
Can anyone explain this to me with rigor?
Thanks.

The fact that $\sin x\to 0$ is easily seen geometrically if you know the statement: "The shortest curve from the point $(x,y)$ to the $x$ axis is of length $|y|$."
This can be used to show that $|\sin x|\leq |x|$, since the arc of the circle from $(\cos x,\sin x)$ to the point $(1,0)$ if length $|x|$ is a path to the $x$-axis, and that has to be at larger than $|\sin x|$. This in turn shows that $\sin x\to 0$ as $x\to 0$.
For $\cos x$, use $\cos^2 x + \sin^2 x = 1$ and that $\cos x$ is positive for $x$ small. Since $\sin x\to 0$, we have $\cos ^2 x\to 1$. Since $\cos x$ is positive for small $x$, this means that $\cos x\to 1$.
