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For what value of p can the function
f(x) = { ((4^x - 1)³)/(sin x · [log|1 + x²/3|]^p), x ≠ 0; 12(log 4)³, x = 0 }
be made continuous at x = 0?
- 1
- 2
- 3
- None of these
Correct answer: 1
Solution
Near 0: (4^x-1)^3 ~ x^3 (ln4)^3, sin(scaled x) ~ x, and [log(1+x^2/3)]^p ~ (x^2/3)^p ~ x^(2p). For a finite nonzero limit the powers of x must balance: 3 = 1 + 2p, giving p = 1.
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