First slide

Continuity

Question

If $f (x) = {\begin{matrix} \frac{6^{x} - 3^{x} - 2^{x} + 1}{2 \sin^{2} (\frac{x}{2})}, & if & x \neq 0 \\ K, & if & x = 0 \end{matrix}$ is continuous at $x = 0,$ then $K =$

Moderate

A

$log (5)$
B

$2 \log 3. \log 2$
C

$32 \log 3. \log 2$
D

$\log 3. \log 2$

Solution

$K = f (0) = \underset{x \to 0}{L t} f (x) = \underset{x \to 0}{L t} \frac{(3^{x} - 1) (2^{x} - 1)}{2 \sin^{2} (\frac{x}{2})} = \underset{x \to 0}{L t} \frac{\frac{(3^{x} - 1) (2^{x} - 1)}{x^{2}}}{2 \frac{\sin^{2} (\frac{x}{2})}{x^{2}}} = \frac{\log 3 . \log 2}{2 . \frac{1}{4}} = 2 \log 3 . \log 2$

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