# List of rules and laws for calculating limits

While looking through our Analysis Script, I got suddenly aware that we had already learned quite an amount of rules and laws for calculating limits.

We got:

• $\displaystyle \lim_{x \to a}\;c = c$
• $\displaystyle \lim_{x \to a}\;x = a$

If the limits exist, one gets with corresponding limits $u,v$

• $\displaystyle \lim_{x \to a} \; f(x) \pm g(x) = u \pm v \quad$
• $\displaystyle \lim_{x \to a} \; cg(x) = cv$
• $\displaystyle \lim_{x \to a} \; f(x) g(x) = uv$
• $\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{u}{v} \quad$ if $v \neq 0$

If the limits exist and if $f(x)$ is continuous:

• $\displaystyle \lim_{x \to a} \; f(x) = f(a)$
• $\displaystyle \lim_{x \to a} \; f(x) \pm g(x) = f(a) \pm g(a)$
• $\displaystyle \lim_{x \to a} \; cf(x) = c f(a)$
• $\displaystyle \lim_{x \to a} \; f(x)g(x) = f(a)g(a)$
• $\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{f(a)}{g(a)} \quad$ if $g(a) \neq 0$

If $f$ is continuous and $\lim\limits_{x \to a} \; g(x)$ exists and if $f(x)$ is defined at $\lim\limits_{x \to a}g(x)$

$$\lim\limits_{x \to a}\; f(g(x)) = f(\lim\limits_{x \to a}\;g(x))$$

As rules we have:

$\forall x$ in an open interval, containing $a$:
$$u(x)\leq v(x)\leq w(x) \implies \lim\limits_{x \to a}\; u(x) \leq \lim\limits_{x \to a}\; v(x) \leq \lim\limits_{x \to a}\; w(x)$$

if $f(x)$ and $g(x)$ is differentiable and if $\lim\limits_{x \to a}\; f(x),g(x) = \pm \infty$ or $\lim\limits_{x \to a}\; f(x),g(x) = 0$ and if $\lim\limits_{x \to a} \frac{f'(x)}{g'(x)}$ exists $$\lim\limits_{x \to a}\frac{f(x)}{g(x)} = \lim\limits_{x \to a} \frac{f'(x)}{g'(x)}$$

But there is definitely more to it, isn't there? For example, when am I allowed to pull a $\lim$ into an infinite sum? I bet Math is simply not ending here. What other rules and laws for calculating limits are there?

EDIT Updated the rules based on the comments.

Your "rules" 3 through 7 are all incorrect; you are assuming that the limit not only exists, but also that the functions $f$ and $g$ are continuous at $a$. The limit may very well exist but not be the value of the function at $a$; the value may be something entirely different, or not even be defined. – Arturo Magidin Jan 24 '11 at 21:24
Your paraphrase of L'Hopital's Rule is also incorrect; you must require the limit of $f'(x)/g'(x)$ to exist; try your "rule" with $f(x) = 6x+\sin x$, $g(x) = 2x+\sin x$, as $x\to\infty$. – Arturo Magidin Jan 24 '11 at 21:27
I think Qiaochu pretty much nailed the issue; I note, however, that you still got the "continuity rule" wrong; your rules 4, 6, and 7 are still incorrect because you are making no assumption about $g$, and that you have now excluded several standard "rules" (if each of two limits exist, then the limit of the sum is the sum of the limits; this is broader than the case of continuity). This suggests that you are focusing on memorization rather than understanding, and the difficulty in memorization is leading you to incorrect statements. – Arturo Magidin Jan 25 '11 at 19:23