# Evaluating limits with fractions

Consider $f(x)=x$ and $g(x)=1/x$. If we take the limit of $f(x)g(x)$ with x tending to $\infty$, why is the limit 1?

At $x=0$, $g(x)$ is undefined, so why should we be allowed to "cancel" out the x in the numerator and denominator? More generally, is $h(x)=f(x)*g(x)=1$? If yes, why?

Note that $$f(x)g(x)=\begin{cases} 1& \text{if}\quad x\neq0\\ \text{undefined}& \text{if}\quad x=0 \end{cases}$$ In particular, given $\epsilon>0$, if $x>1$, then $|f(x)g(x)-1|=0<\epsilon$ . Thus by the definition of the limit $$\lim_{x\to\infty} f(x)g(x)=1$$ Intuitively to take a limit as $x\to \infty$ we only need to know the values of the function above some threshold $M$. What happens below $M$ is irrelevant for the purposes of taking the limit
For values of $x$ where $x\ne0$, $f(x)*g(x)$ will clearly be exactly 1, because for those values there are no problems at all with it being undefined. For example, if $x=5$, then $f(x)*g(x)={5\over 5}$, which is clearly just 1.
As you take the limit to infinity, you can "cancel out the x in the numerator and the demoninator", because for those values that are not zero, the function is very clearly defined. For the special case where $x=0$, you are correct in saying the x's cannot be cancelled out.
$h(x)$ is not defined at $0$ because $g(x)$ is not defined at zero. This, however, does not at all impact our ability to take the limit as $x\to \infty$.