Answer
The function $f(x)$ is continuous at $c=5$.
Work Step by Step
We are given:
$f(x)=3x^4-x^2+2$
We need to determine if the function $f(x)$ is continuous at $c=5$.
We check the left-hand and right-hand limits. If they are equal to each other and the value of the function at the limit, then the function is continuous at that point.
$\lim\limits_{x \to 5^{-}}f(x)=\lim\limits_{x \to 5^{-}} 3x^4-x^2+2 \\=\lim\limits_{x \to 5^{-}}(3x^4)-\lim\limits_{x \to 5^{-}} x^2+2 \\=(3)(5^4)-(5^2)+2\\=1852$
$\lim\limits_{x \to 5^{+}}f(x)=\lim\limits_{x \to 5^{+}} 3x^4-x^2+2 \\=\lim\limits_{x \to 5^{+}}(3x^4)-\lim\limits_{x \to 5^{+}} x^2+2 \\=(3)(5^4)-(5^2)+2\\=1852$
$\lim\limits_{x \to 5^{-}}f(x)=\lim\limits_{x \to 5^{+}}f(x)=f(5)$
Therefore, the function $f(x)$ is continuous at $c=5$.
