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Question
Discuss the continuity of the following function at the point indicated against them :
f(x) `{:(=("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "for" x ≠ 0),(= 1",", "for" x = 0):}}` at x = 0
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Solution
f(0) = 1 ...(Given)
`lim_(x -> 0^-) "f"(x) = lim_("h" -> 0) ("e"^(1/((0 - "h"))) - 1)/("e"^(1/((0 - "h"))) + 1)`
= `lim_("h" -> 0) ("e"^((-1)/"h" - 1))/("e"^((-1)/"h" + 1))`
= `lim_("h" -> 0) ("e"^(1/(1/"h")) - 1)/("e"^(1/(1/"h")) + 1)`
= `(0 - 1)/(0 + 1)`
= – 1
`lim_(x -> 0^+) "f"(x) = lim_("h" -> 0) ("e"^(1/((0 + "h"))) - 1)/("e"^(1/((0 + "h"))) + 1)`
= `lim_("h" -> 0) ("e"^(1/"h") - 1)/("e"^(1/"h") + 1)`
= `lim_("h" -> 0) ("e"^(1/"h")(1 - 1/("e"^(1/"h"))))/("e"^(1/"h")(1 + 1/("e"^(1/"h")))`
= `(1 - 0)/(1 + 0)`
= 1
`lim_(x -> 0^-) "f"(x) ≠ lim_(x -> 0^+) "f"(x)`
∴ f(x) is discontinuous at x = 0
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