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Question
Find the points of discontinuity of f, where f(x) = `{(sinx/x", if" x<0),(x + 1", if" x >= 0):}`.
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Solution
f(x) = `{(sinx/x", if" x<0),(x + 1", if" x >= 0):}`
At x = 0, f(0) = 1
L.H.L. = `lim_(x->0^-)` f(x) = `lim_(h->0)(sin(-h))/-h` = 1
R.H.L. = `lim_(x->0^+)` f(x) = `lim_(h->0)` (h + 1) = 0 + 1 = 1
`lim_(x->0^-)` f(x) = `lim_(x->0^+)` f(x) = f(0)
∴ f is continuous at x = 0.
When x < 0, sin x and x are both continuous.
∴ `sinx/x` is also continuous.
When x > 0, f(x) = x + 1 is a polynomial.
∴ f is continuous.
⇒ f is not discontinuous at any point.
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