Determine the value of the constant ‘k’ so that the function is continuous at x = 0.

Question

Determine the value of the constant ‘k’ so that the function $straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell fraction numerator Kx over denominator vertical line straight x vertical line end fraction comma space If space straight x space less than 0 end cell row cell 3 comma space if space straight x greater or equal than space 0 end cell end table close$ is continuous at x = 0.

Solution

straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell fraction numerator Kx over denominator vertical line straight x vertical line end fraction comma space If space straight x space less than 0 end cell row cell 3 comma space if space straight x greater or equal than space 0 end cell end table close stack lim space straight f left parenthesis straight x right parenthesis with straight x space rightwards arrow 0 below space equals stack space lim with straight x rightwards arrow 0 below space straight f left parenthesis space straight x right parenthesis space equals space straight f left parenthesis 0 right parenthesis rightwards double arrow space limit as straight x space rightwards arrow 0 of space fraction numerator negative kx over denominator straight x end fraction space equals space stack lim space 3 with straight x rightwards arrow 0 below space equals space 3 rightwards double arrow negative straight k space equals 3 rightwards double arrow straight k equals negative 3