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Question
If f(x) `{:(= (sin2x)/(5x) - "a"",", "for" x > 0),(= 4 ",", "for" x = 0),(= x^2 + "b" - 3",", "for" x < 0):}}` is continuous at x = 0, find a and b
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Solution
f(x) is continuous at x = 0
∴ `lim_(x -> 0^+) "f"(x)` = f(0)
∴ `lim_(x -> 0^+) ((sin2x)/(5x) - "a")` = 4
∴ `lim_(x -> 0^+) (sin 2x)/(5x) - lim_(x -> 0^+) "a"` = 4
∴ `1/5 lim_(x -> 0^+) (sin2x)/(2x) xx (2) - lim_(x -> 0^+) "a"` = 4
∴ `1/5 (1) (2) - "a"` = 4 ...`[(because x -> 0"," 2x -> 0),(lim_(x -> 0^+) sintheta/theta = 1)]`
∴ `2/5 - "a"` = 4
∴ `2/5 - 4` = a
∴ a = `-18/5`
Also, `lim_(x -> 0^+) "f"(x)` = f(0)
∴ `lim_(x -> 0^+) (x^2 + "b" - 3)` = 4
∴ b – 3 = 4
∴ b = 7
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