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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.

Thanks in advance

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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 don't think it's profitable to think of limits in terms of the laws they satisfy. If you thoroughly understand the definition, the rest will follow, and in particular you really only need the fact that limits are preserved by continuous functions (alternately the defining property of limits or the defining property of continuous functions), and maybe l'Hopital's rule once in awhile. – Qiaochu Yuan Jan 24 '11 at 21:34
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
@ftiaronsem: Still got the continuity limit (your 12th formula) wrong; and now formulas 7 through 11 are easy consequences of formulas 1-6 and formula 12. Really: it's not about memorization, it's about understanding. This exercise should convince you that trying to memorize "all the rules and laws" accurately is not a good way to spend your time. – Arturo Magidin Jan 25 '11 at 19:50

Here is a list of sites for you to explore your question. You are right math is never done with any topic because there always seems to be another way to view the topic. If you really like limits I would suggest giving the book a read. Enjoy. Please, update your list if you find any other rules to post.

  1. TopMath $\Leftarrow$ likely what you want to read
  2. Limit Rule List
  3. Hints on Limits
  4. The Limit Law
  5. Mathworld's Limit page
  6. MathCS real analysis def of Limit
  7. YouTube video on limits in Analysis
  8. Louisville's Chapter 6 (limits) of Real Analysis
  9. Limits: A New Approach to Real Analysis $\Leftarrow$ a book you should read
  10. Complex Limits and Continuity
  11. Limits and Continuity - Complex Analysis
  12. Proof Wiki: Definition of a point
  13. Analysis Webnotes
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