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प्रश्न
Discuss the continuity of the function f at x = 0
If f(x) = `(2^(3x) - 1)/tanx`, for x ≠ 0
= 1, for x = 0
उत्तर
Given f(0) = 1
Consider,
`lim_(x->0)` f (x) = `lim_(x->0) [(2^(3x) - 1)/tanx]`
= `lim_(x->0) [((2^(3x) - 1)/x)/((tanx)/x]], x ≠ 0`
= `lim_(x->0) [(2^(3x) - 1)/(3x).3]/(lim_(x->0)(tanx)/x) = 3 log 2`
= log 8
`(lim_(x->0) (a^x - 1)/x = log a and lim_(x->0) (tan x)/x = 1)`
`as x -> 0, 3x ->0`
Since `(lim_(x->0)` f(x) ≠ f(0)
f(x) is discontinuous at x = 0
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