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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.