Finding a Limit of the Heaviside Function

Oliver Heaviside (1850–1925) was a self-taught mathematician and electrical engineer. He developed a branch of mathematics called operational calculus in which differential equations are solved by converting them to algebraic equations. Heaviside applied vector calculus to electrical engineering and, perhaps most significantly, he simplified Maxwell's equations to the form used by electrical engineers to this day. In 1902 Heaviside claimed there is a layer surrounding Earth from which radio signals bounce, allowing the signals to travel around the Earth. Heaviside's claim was proved true in 1923. The layer, contained in the ionosphere, is named the Heaviside layer. The function we discuss here is one of his minor contributions to mathematics and electrical engineering.

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.