Answer
Prove that $\lim\limits_{x\to2}f(x)=f(2)$
Work Step by Step
*NOTES TO REMEMBER: $f(x)$ is continuous at $a$ if and only if $$\lim\limits_{x\to a}f(x)=f(a)$$
We consider
$\lim\limits_{x\to2}f(x)$
$=\lim\limits_{x\to2}(3x^4-5x+\sqrt[3]{x^2+4})$
$=3\lim\limits_{x\to2}x^4-5\lim\limits_{x\to2}x+\lim\limits_{x\to2}\sqrt[3]{x^2+4}$
$=3\lim\limits_{x\to2}x^4-5\lim\limits_{x\to2}x+\sqrt[3]{\lim\limits_{x\to2}(x^2+4)}$
$=3\lim\limits_{x\to2}x^4-5\lim\limits_{x\to2}x+\sqrt[3]{\lim\limits_{x\to2}x^2+\lim\limits_{x\to2}4}$
$=3\times2^4-5\times2+\sqrt[3]{2^2+4}$
$=f(2)$
Therefore, $f(x)$ is continuous at $2$.
