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Question
Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if" x ≠ 0),(0",", "if" x = 0):}` is discontinuous at x = 0.
Solution
The left hand limit of f at x = 0 is given by
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) ("e"^(1/x) - 1)/("e"^(1/x) + 1)`
= `(0 - 1)/(0 + 1)`
= −1
Similarly, `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) ("e"^(1/x) - 1)/("e"^(1/x) + 1)`
= `lim_(x -> 0^+) (1 - 1/"e"^(1/x))/(1 + 1/"e"^(1/x))`
= `lim_(x -> 0^+) (1 - "e"^((-1)/x))/(1 + "e^((-1)/x)`
= `(1 - 0)/(1 + 0)`
= 1
Thus `lim_(x -> 0^-) "f"(x) ≠ lim "f"(x)_(x -> 0^+)`
Therefore, `lim_(x -> 0) "f"(x)` does not exist. Hence f is discontinuous at x = 0.
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