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