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