measure the speed of convergence of two functions

Question

Let's say I have two functions $f$ and $g$ such that :

$$\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = l$$

Then I can say that : $$f \sim_a g$$

So when I say that two functions are equivalent near a point, it just basically means that they converge to the same value (when the value is not infinity or $0$).

For example, let's take : $f(x) = a^x$ and $g(x) = a^{x^2}$ where $a \in ]0,1]$. Then we obviously have :

$$\lim_{x \to 0} f(x) = \lim_{x \to 0} g(x) = 1$$

So, I can say that :

$$f \sim_0 g$$

But my problem here is that these two functions don't converge to $1$ at the same speed. $x^2$ tend to $0$ faster than $x$.

So how can I measure the speed ratio of two functions near a point? Is there a tool that allows me to say: $g$ tends to $1$ faster than $f$.

The reason why I am asking this question is the following :

If I want to calculate the value of the following limit :

$$\lim_{h \to 0} \sum_{n = 0}^{\infty} (-1)^nf(nh)$$

Where $f(x) = a^{x}$, then saying that : $f(nh) \sim_{h \to 0} g(nh)$ where $g(x) = a^{x^2}$ doesn't necessaraly imply that :

$$\lim_{h \to 0} \sum_{n = 0}^{\infty} (-1)^nf(nh) = \lim_{h \to 0} \sum_{n = 0}^{\infty} (-1)^ng(nh)$$ right ?

So to me in order to say that the two above limits are equal, we must prove that $f$ and $g$ converge at the same speed, that's the reason why I want a tool that allows me to say: these two functions converge at the same speed hence the limit above are equal.

I hope everything is clear. If I said something wrong don't hesitate to correct me.

Answer 1

If $f$ and $g$ are two functions such that $\lim_{x->a} f(x)= \lim_{x\to a}g(x)=c \in \mathbb{R}$.

• The function $f$ goes faster to $c$ than $g$ for $x\to a$ if $$ \lim_{x\to a} \frac{f(x)-c}{g(x)-c} =0.$$
• The function $f$ goes slower to $c$ than $g$ for $x\to a$ if $$ \lim_{x\to a} \frac{f(x)-c}{g(x)-c} =\pm \infty.$$
• The function $f$ goes equally fast to $c$ as $g$ for $x\to a$ if $$ \lim_{x\to a} \frac{f(x)-c}{g(x)-c} \in \mathbb{R}\setminus\{0\}.$$

Two remarks: (1) Similar definitions can be given for two sequences $\{a_n\}$ and $\{b_n\}$ converging to a same real number as $n\to\infty$.

(2) When $f$ and $g$ both diverge to $\infty$ (when $x\to \infty$ for instance) ones says that $f$ goes faster to infinity than $g$ is $$ \lim_{x\to\infty} \frac{g(x)}{f(x)} = 0.$$ Note that now $f$ is in the denominator, not in the nominator.