I'm Looking for an advanced textbook on limits . There is an alternative way to finding limits which I couldnt find anywhere on the internet or in any of my Books about Calculus that specificly teaches that and prooves the formulas.
Simple Example : Sin(x) acts like y = x near zero and thats proved by linear approximation Or simply by looking at the graph.
A more compliacted Example which I'm looking for a proof:
when $$\lim_{x\to \infty} \sqrt{ax^n+bx^{n-1}+...} $$ is equvalent to :
$$\sqrt{n} {.}|{x+\frac{b}{na}}| $$
and there are a lot of formulas like this which I like to learn very carefully and efficient.
We call this things equivalence relations of limits in our country and we show it with "~"
Another one :
when $$\lim_{x\to \infty} {1^k + 2^k + 3^k + 4^k + ... + n^k} $$
is equivalent to :
$$ \frac{1}{k+1} {.} {n^{k+1}} $$
I'd appreciate It if someone could help me find a good resource to this approach of limit finding which is really fast sometimes.