calculus - Continuity of polynomials using sequence definition of continuity.

The three step continuity definition states that :

$f(x_0)$ exists;

$\lim_{x \to x_o} f(x)$ exists; and

$\lim_{x \to x_o} f(x)$ =$f(x_o)$.

can we use this definition to prove that $f(x)=a_0+a_1x+a_2x^2+\ldots+a_nx^n$ is continuous at every point $x \in \mathbb R$.
If not, then how can we prove it. Please help$\ldots$

Answer

Yes you can use it but you should also apply limit laws since it is summation of individual functions i.e., $x , x^2....$ and so on.

here is the link for the proof https://proofwiki.org/wiki/Polynomial_is_Continuous