# measure the speed of convergence of two functions

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.

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.