# What inequalities should one know to evaluate limits fluently?

During the Calculus course, we often used common inequalities to estimate the terms of a sequence and find its limit in the end. The problem is that these inequalities, obvious though they may be, seldom come to mind if you have not used them to solve similar problems at least once. I have listed some of them but I think there are more - what should I add to the list?

1. Bernoulli's inequality
2. $n! > 2^n \iff n \ge 4$
3. $2^n > n^2\iff n \ge5$
• i wonder why Cauchy-Schwarz didn't show up anywhere... also Hölder is sometimes pretty useful. – tired Oct 27 '17 at 16:59
• Muirhead as well. – mtheorylord Oct 27 '17 at 19:28
• FYI it's not just in calculus. It also comes up a lot when you want to analyze an error or other kind of deviation probabilistically (e.g. $\epsilon$ deviation with $\delta$ probability). – Mehrdad Oct 27 '17 at 20:02

$-1\leq \sin(x)\leq 1$, and the same is true for $\cos$

The AM-GM inequality is so popular it has its own tag on this site.

This inequality involving the choose function is pretty commonly used when the choose function or exponential function pop up $$\frac{n^k}{k^k}\leq {n\choose k}\leq\frac{n^k}{k!}\leq \frac{(n\cdot e)^k}{k^k}$$

• $|\sin(x)| \leq |x|$ is also handy. – Bungo Oct 27 '17 at 18:29

A good thing to have in mind is the hierarchy of common functions. In synthetic form

For all $a∈ℝ$, $b⩾1$, $c>1$, $d>1$ and $x>1$

$$a ≺ \log(\log(n)) ≺ \log(n)^b ≺ n^{\frac{1}{c}} ≺ n ≺ n \log(n) ≺ n^d ≺ x^n ≺ n!≺ n^n$$

when $n⟶+∞$, using Hardy's notations for asymptotic domination¹.

Note that this does not give you the specific $N$ where one gets superior to the other, though.

You might also derive useful comparisons from the Théorème de croissances comparées for which I don't know of any equivalent in English literature, but which is simply the following

For all $a>0$ and $b>0$ $$x^a = o_{+∞}(e^{bx})$$ $$\ln(x)^a = o_{+∞}(x^b)$$ and $$\lvert\ln(x)\rvert^a = o_{0}(x^b)$$

1. Which is messed up, because those are obsolete, but there are not other convenient notations for asymptotic domination as a (strict partial) order. What I was taught was that the Vinogradov notation $f ≪ g$ stood for $f=o(g)$, but apparently it is $f=O(g)$ instead. We really need a standardization of those things.

Two useful things to know in calculating limits are:

• For all $a>1,\alpha > 0$, there exists some $N$ such that $\log_a n < n^\alpha$ for $n>N$
• For all $a>1, \alpha > 0$, there exists some $N$ such that $a^n > n^\alpha$ for all $n>N$

In other words, the exponential function is faster than any power, and log is slower than any power.

• You're second bullet requires $a > 1$, correct? Otherwise you have a decaying exponential. – User8128 Oct 28 '17 at 3:54

For any function such that

$$x\in(a,b)\implies f''(x)>0$$

then

$$c,x\in[a,b]\implies f(c)+f'(c)(x-c)\le f(x)$$

$$c,d\in[a,b],~x\in[c,d]\implies f(c)+\frac{f(c)-f(d)}{c-d}(x-c)\ge f(x)$$

Most clear if you graph it out. All inequalities flipped when $f''(x)<0$.

For example,

$$x\in(-\infty,\infty)\implies1+x\le e^x$$

$$x\in[0,\pi]\implies x\ge\sin(x)$$

A lot of common inequalities can be derived from this. (Though note you have to take the derivative, so don't use it to prove fundamental derivatives.)

• In words: The graph of a convex function lies above its tangent lines. – Martin R Oct 28 '17 at 3:12
• @MartinR And below its secant lines. – Simply Beautiful Art Oct 28 '17 at 19:10

In general one should already be aware of the more famous and elementary inequalities like AM-GM. You have also noted down the Bernoulli inequality which is much simpler, less famous and highly powerful and useful in unexpected ways. Apart from the famous ones (those with a name) I find the following very helpful:

• $\sin x<x<\tan x$ for $x\in(0,\pi/2)$.
• $\log x\leq x-1$ for $x>0$.
• $e^{x} \geq 1+x$ for all real $x$.