f(x)=(36)x−9x−4x+12sin2⁡(x/4) if x≠0k if x=0continuous at x=0 then k=

k=Ltx→0(36)x−9x−4x+12sin2⁡(x/4)=Ltx→09x−14x−12sin2⁡(x/4)=(log⁡9)(log⁡4)2X116=8X2log⁡3⋅X2log⁡2=32log⁡2⋅log⁡3

f(x)=(36)x−9x−4x+12sin2⁡(x/4) if x≠0k if x=0continuous at x=0 then k=

k=Ltx→0(36)x−9x−4x+12sin2⁡(x/4)=Ltx→09x−14x−12sin2⁡(x/4)=(log⁡9)(log⁡4)2X116=8X2log⁡3⋅X2log⁡2=32log⁡2⋅log⁡3

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$f (x) = \{\begin{array}{cc} \frac{(36)^{x} - 9^{x} - 4^{x} + 1}{2 \sin^{2} (x / 4)} & if x \neq 0 \\ k & if x = 0 \end{array}$ continuous at $x = 0$ then $k =$

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$32 \log 2 \cdot \log 3$

$8 \log 5$

$8 \log (6)$

$\log 2 \cdot \log 3$

answer is C.

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Detailed Solution

$\begin{array}{l} k = \underset{x \to 0}{Lt} \frac{(36)^{x} - 9^{x} - 4^{x} + 1}{2 \sin^{2} (x / 4)} \\ = \underset{x \to 0}{Lt} \frac{(9^{x} - 1) (4^{x} - 1)}{2 \sin^{2} (x / 4)} \\ = \frac{(\log 9) (\log 4)}{2 X \frac{1}{16}} = 8 X 2 \log 3 \cdot X 2 \log 2 \\ = 32 \log 2 \cdot \log 3 \end{array}$