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A polynomial can simply use direct substitution.

$\lim_{x\to0}3x^2-2x+1=\lim_{x\to 1}(3\times0^2-2\times0+1)=1$

A rational function can easily use direct substitution on its domain.

$\lim_{x\to0}\frac{1-x^2}{1+x}=1$

Does an algebraic function can easily use direct substitution on its domain?

$\lim_{x\to0}\sqrt[3]{x^2}$

In this limit, I can use continuity to make the limit under the triple square root, and use direct substitution on the polynomial $x^2$.

$\lim_{x\to0}\sqrt[3]{x^2}=\sqrt[3]{\lim_{x\to0}x^2}=\sqrt[3]{0^2}=0$.

But the direct substitution property just say,

If $f$ is a polynomial or a rational function and $a$ is in the domain of $f$, then $\lim_{x\to a}f(x)=f(a)$.

Why not just simply say 'if $f$ is an algebraic function...'?

Is there are any pitfall that I have not realized?

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    Not only does this hold for an "algebraic function", but for any continuous function. – Anne Bauval Dec 15 '22 at 15:41
  • @AnneBauval I agree with your comment, however it may be that OP meant to ask whether all algebraic functions are continuous on their domains, so that one knows a function is algebraic, one can find its limit as $x \to c$ by "just plugging in" after checking if $c$ is in its domain. – coffeemath Dec 15 '22 at 15:50
  • There are plenty of pitfalls here. The simplest case is that of the square root, whose real domain is the nonnegative numbers, but whose complex domain can extend to any “simply-connected” set not containing the origin. Try to define this function in any (complex) neighborhood of the origin and you get hung up on the fact that every complex number has two square roots, in most cases with no reason to choose one over the other. But once your function is truly well-defined, you can go ahead with your substitution process. – Lubin Dec 15 '22 at 18:16
  • There is a huge class of functions where plugging works for evaluation of limits. See more details at https://math.stackexchange.com/a/1822706/72031 – Paramanand Singh Jan 12 '23 at 02:29
  • thanks,now i understand why is direct substitution property defined like this –  May 29 '23 at 14:31

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The limit of a function $f(x)$ at $x=a$ is defined to be:

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

The limit definition is generalized for a lot of fields in math.

algebraic functions are continuous for their domain, therefore if one knows a function is algebraic and that $a$ is in its domain, he can "just" calculate the value of $f(a)$ without the use of limit.

knight5478
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