Is there some trick as in to seeing what a franctioned function looks like, when you know how the denominator and the numerator look? For example I can instantly plot $\sin(x)$ in my head. I can also instantly plot $x$ in my head. But I have no idea on how to do this with $\displaystyle\frac{\sin(x)}{x}$.
It would be handy if I knew a trick, for example to quickly see why $\displaystyle\lim_{x \to 0}\frac{\sin(x)}{x} = 1$.
It would also be handy for application in the big O notation where $f(x) = O(g(x)) \iff \displaystyle\lim_{x \to \infty}\ \frac{f(x)}{g(x)} =c$. I am currently having a hard time using this defition because of my struggle with fractioned functions.
Around $x = 0$, $\sin x$ looks like $x$. Can you visualize this? –  Qiaochu Yuan Dec 8 '11 at 0:07
Because $\sin(x) = x$ for very small $x$? Yes. I hope you are not trying to mock me with "can you visualize this" though. The key of my question is figuring out a general way of looking at fractioned functions, not validating $\displaystyle\lim_{x \to 0}\frac{\sin(x)}{x} = 1$ by the way. But perhaps your comment is going to go further? –  user12205 Dec 8 '11 at 0:15
In general it's hard. But it is not too hard to visualize how the standard sorts of functions behave when $x$ is large, and their ratios can also be handled from informal things like $a^x$ (for $a>0)$ grows faster than any polynomial. So one can then "see" what happens to $\frac{x^3-3x^2-77}{1.2^x}$ as $x$ gets large. –  André Nicolas Dec 8 '11 at 0:46
For $P(x)\sin x$, you can think of the polynomial $P(x)$ as an "envelope" modifying the amplitude of $\sin$. For a rational function (ratio of two nonzero polynomials), you can get a good amount of information from the roots. The roots of the numerator give you the roots of the function, the roots of the denominator given you asymptotes. It can be a little more tricky than that: tutorial.math.lamar.edu/Classes/Alg/GraphRationalFcns.aspx –  dls Dec 8 '11 at 0:47
@Jos: I had no intention of mocking you. I just wanted to mention that I find it easier to visualize the fact that $\sin x$ is approximately $x$ for $x$ small (it is not strictly speaking correct to say that they are equal) than to visualize the entire graph of $\frac{\sin x}{x}$, which contains a lot of extraneous information that isn't relevant to determining the behavior of the function near $0$. –  Qiaochu Yuan Dec 8 '11 at 1:05