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Question
Discuss the continuity of the following function at the point indicated against them :
f(x) `{:(=(4^x - 2^(x + 1) + 1)/(1 - cos 2x)",", "for" x ≠ 0),(= (log 2)^2/2",", "for" x = 0):}}` at x = 0.
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Solution
f(0) = `(log 2)^2/2` ...(Given) ...(1)
`lim_(x -> 0) "f"(x) (4^x - 2^(x + 1) + 1)/(1 - cos 2x)`
= `lim_(x -> 0) ((2^x)^2 - 2(2^x) + 1)/(2sin^2x)`
= `lim_(x -> 0) (2^x - 1)^2/(2sin^2x)`
= `lim_(x -> 0) ([(2^x - 1)^2/x^2])/(2((sin^2x)/x^2)` ...[∵ x → 0, x ≠ 0 ∴ x2 ≠ 0]
= `1/2 [lim_(x -> 0) (2^x - 1)/x]^2/[lim_(x -> 0) sinx/x]^2`
= `1/2 * (log 2)^2/(1)^2 ...[because lim_(x -> 0) ("a"^x - 1)/x = log "a"]`
= `(log 2)^2/2`
From (1) and (2)
`lim_(x -> 0) "f"(x)` = f(0)
∴ f is continuous at x = 0
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