Continuity at a Point II

Problem 233: Determine whether the following functions are continuous at $a$ .

(1) $\begin{equation*} f(x) = \frac{3x^2 + 2x + 1}{x-1}, \quad \text{at} \quad a = 2\end{equation*}$

Solution: Before we determine whether the given function is continuous or not, we need to know when is a function continuous. That is,

Theorem 1: A function $f$ is continuous at $a$ if

(2) $\begin{equation*} \lim_{x \to a} f(x) = f(a). \end{equation*}$

Using Theorem 1, one can now determine whether (1) is continuous or not. Notice that

(3) $\begin{equation*} f(2) = \frac{3(2)^2 + 2(2) + 1}{2-1}=\frac{17}{1}=17.\end{equation*}$

Moreover,

(4) $\begin{equation*} \lim_{x \to 2} g(x) = \lim_{x \to 2} \frac{3x^2 + 2x + 1}{x-1} = \frac{3(2)^2 + 2(2) + 1}{2-1} = 17.\end{equation*}$

Thus, because $g$ is a rational function and the denominator is nonzero at 2, it follows by Theorem 1 that $\lim_{x \to 2} g(x) = g(2)=17$ . Therefore, $g$ is continuous at 2.

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