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Question
Examine whether the function is continuous at the points indicated against them :
f(x) `{:(= x/(tan3x) + 2",", "for" x < 0),(= 7/3",", "for" x ≥ 0):}} "at" x = 0`
Solution
`lim_(x -> 0^+) "f"(x) = 7/3` ...(given)
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) (x/(tan3x) + 2)`
= `lim_(x -> 0^-) x/(tan 3x) + lim_(x -> 0^-) 2`
= `lim_(x -> 0^-) 1/((tan3x)/x) + lim_(x -> 0^-) 2`
= `lim_(x -> 0^-) (1/((tan3x)/(3x) xx 3)) + lim_(x -> 0^-) 2`
= `(lim_(x -> 0^-) 1)/(3 lim_(x -> 0^-) ((tan 3x)/(3x))) + lim_(x -> 0^-) 2`
= `1/(3(1)) + 2`
= `7/3`
∴ `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^-) "f"(x)`
∴ f(x) is continuous at x = 0.
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