Question

Which of the following functions is not continuous on domain? a) |x-1| b) √x c) 1/x²+2x+3 d) f(x) = cos x for x not = 0 and f(x) = 1 for x = 0

Answer 1

Function f is defined at x=-2 since

i.) f(-2) = (-2)2 + 2(-2) = 4-4 = 0 .

The left-hand limit

$ \displaystyle{ \lim_{ x \to -2^{-} } f(x) = \lim_{ x \to -2^{-} } (x^2 + 2x) } $

= (-2)2 + 2(-2)

= 4 - 4

= 0 .

The right-hand limit

$ \displaystyle{ \lim_{ x \to -2^{+} } f(x) = \lim_{ x \to -2^{+} } (x^3 - 6x) } $

= (-2)3 - 6(-2)

= -8 + 12

= 4 .

Since the left- and right-hand limits are not equal, ,