Find \(\lim\limits_{t\rightarrow 0}u_{0}(t) ,\) where \(u_{0}(t) =\left\{ \begin{array}{l@{\quad}ll} 0 & \hbox{if} & t\lt 0 \\ 1 & \hbox{if} & t\geq 0 \end{array} \right. \)
Solution Since this Heaviside function changes rules at \(t=0\), we find the one-sided limits as \(t\) approaches \(0\). \[ \hbox{For \(t\lt 0\), \(\lim\limits_{t\rightarrow 0^{-}}u_{0}(t) =\lim\limits_{t\rightarrow 0^{-}}0=0\) and for \(t\geq 0\), \(\lim\limits_{t\rightarrow 0^{+}} u_{0}( t) =\lim\limits_{t\rightarrow 0^{+}} 1=1\)} \]
Since the one-sided limits as \(t\) approaches \(0\) are not equal, \(\lim\limits_{t\rightarrow 0}\) \(u_{0}(t) \) does not exist.