Question

The Signum Function The signum (or sign) function, denoted by sgn, is defined by

\({\bf{sgn}}x = \left\{ {\begin{array}{*{20}{c}}{ - {\bf{1}}}&{{\bf{if}}\,\,x < {\bf{0}}}\\{\bf{0}}&{\,{\bf{if}}\,\,x = {\bf{0}}}\\{\bf{1}}&{\,\,\,{\bf{if}}\,\,x > {\bf{0}}}\end{array}} \right.\)

(a) Sketch the graph of this function.

(b) Find each of the following limits or explain why it does not exist.

(i) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{0}}^ + }} {\bf{sgn}}x\) (ii) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{0}}^ - }} {\bf{sgn}}x\)

(iii) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{0}}} {\bf{sgn}}x\) (iv) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{0}}} \left| {{\bf{sgn}}x} \right|\)

Short answer

(a)

(b) (i) 1 (ii) \( - 1\)

(iii) does not exist (iv) 1

Step-by-step solution

Skecth the graph of signum function

The figure below represents the graph of the signum function.

Find the value of limits

The value of the expression \(\mathop {\lim }\limits_{x \to {0^ + }} {\mathop{\rm sgn}} x\)(right-hand limit)can be calculated as,

\(\mathop {\lim }\limits_{x \to {0^ + }} {\mathop{\rm sgn}} x = 1\)

The value of the expression \(\mathop {\lim }\limits_{x \to {0^ - }} {\mathop{\rm sgn}} x\)(left-hand limit) can be calculated as,

\(\mathop {\lim }\limits_{x \to {0^ - }} {\mathop{\rm sgn}} x = - 1\)

As the left hand limit and right hand limit are not equal, therefore the expression \(\mathop {\lim }\limits_{x \to 0} {\mathop{\rm sgn}} x\) is not defined.

The value of the expression \(\mathop {\lim }\limits_{x \to 0} \left| {{\mathop{\rm sgn}} x} \right|\) can be calculated as,

\(\begin{array}{c}\mathop {\lim }\limits_{x \to 0} \left| {{\mathop{\rm sgn}} x} \right| &=& \mathop {\lim }\limits_{x \to 0} \left( 1 \right)\\ &=& 1\end{array}\)