Determining Whether a Function Is Continuous at a Number

1. Determine whether \(f(x)=3x^{2}-5x+4\) is continuous at \(1\).
2. Determine whether \(g(x)=\dfrac{x^{2}+9}{x^{2}-4}\) is continuous at \(2\).

Solution (a) We begin by checking the conditions for continuity. First, \(1\) is in the domain of \(f\) and \(f( 1) =2.\) Second, \(\lim\limits_{x\rightarrow 1}f(x)=\lim\limits_{x\rightarrow 1}(3x^{2}-5x+4)=2,\) so \(\lim\limits_{x\rightarrow 1}f( x) \) exists. Third, \(\lim\limits_{x\rightarrow 1}f( x) =f( 1)\). Since the three conditions are met, \(f\) is continuous at \(1\).

(b) Since \(2\) is not in the domain of \(g\), the function \(g\) is discontinuous at \(2\).