Equal Functions

Topics: Calculus - Taylor Polynomial

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(definition)

Let $f,g:\mathbb{R}\to \mathbb{R}$. We say that these two functions are equal in $a\in \mathbb{R}$ up to $n$ order if:

$$\underset{x\to a}{\lim }\frac{f(x)-g(x)}{(x-a{)}^n}=0$$

(theorem)

Let $f:\mathbb{R}\to \mathbb{R}$ be differentiable $n$ times with a continuous derivative in $a\in \mathbb{R}$.

Then, $f$ and $P_f(n,a)$ (its Taylor polynomial) are equal up to the $n$ order.

(theorem)

If $p(x)$ and $q(x)$ are degree $n$ polynomials and they’re equal in $a\in \mathbb{R}$ up to $n$ order, then they’re actually the same polynomial.