Question

f(x) = `{{:(|x - 4|/(2(x - 4))",", "if"  x ≠ 4),(0",", "if"  x = 4):}` at x = 4

Answer 1

We have, f(x) = `{{:(|x - 4|/(2(x - 4))",", "if"  x ≠ 4),(0",", "if"  x = 4):}`

At x = 4

L.H.L. = `lim_(x -> 4^-) |x - 4|/(2(x - 4))`

= `lim_("h" -> 0) (|(4 - "h") - 4|)/(2[(4 - "h") - 4])`

= `lim_("h" -> 0) |-"h"|/(-2"h")`

= `lim_("h" -> 0) |"h"|/(-2"h")`

= `lim_("h" -> 0) "h"/(-2"h")`

= `(-1)/2`

But given that f(4) = 0 ≠ L.H.L.

So, f(x) is discontinuous at x = 4.