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Question
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)",", "for" x ≠ 2),(= 1/5",", "for" x = 2):}}`at x = 2
Solution
f(2) = `1/5` ...(Given) ...(1)
`lim_(x -> 2) "f"(x) = lim_(x -> 2) (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)`
Put x = 1 + h. Then as x → 2, h → 1
Also x – 1 = h and x – 2 = (1 + h) – 2 = h – 1
∴ `lim_(x -> 2) "f"(x) = lim_("h" -> 1) (sqrt("h") - "h"^(1/3))/("h" - 1)`
= `lim_("h" -> 1) ((sqrt("h") - 1) - ("h"^(1/3) - 1))/("h" - 1)`
= `lim_("h" -> 1) [(sqrt("h") - 1)/("h" - 1) - ("h"^(1/3) - 1)/("h" - 1)]`
= `lim_("h" -> 1) (("h"^(1/2) - 1^(1/2))/("h" - 1)) - lim_("h" -> 1) (("h"^(1/3) - 1^(1/3))/("h" - 1))`
= `1/2(1) ^(-1/2) - 1/3(1)^(-2/3) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `1/2 - 1/3`
= `1/6`
From (1) and (2),
`lim_(x -> 2) "f"(x) ≠ "f"(2)`
∴ f is discontinuous at x = 2
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