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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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    The condition *if $f(x)$ is defined at $\lim\limits_{x\to a}g(x)$* should read *if $f$ is defined in$a$neighborhood of $\lim\limits_{x\to a}g(x)$* (two modifications).2011-11-10

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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