Question

$\underset{x\to 0}{lim}\frac{1-cos3x}{x(3^x-1)}=$

• A. $\frac{9}{2}$
• B. $\frac{9}{\left(2\log 3\right)}$
• C. $\frac{9\log 3}{2}$
• D. $1$

Answer 1

The correct option is B $\frac{9}{\left(2\log 3\right)}$

$\underset{x\to 0}{lim}\frac{1-\cos 3x}{x(3^x-1)}$

$cos3x=1-\frac{9x^2}{2!}+\dots$

$1-cos3x=\frac{9x^2}{2!}+\dots$ ......... (1)

$\underset{x\to 0}{lim}\frac{3^x-1}{x}=log3$ .......... (2)

$\underset{x\to 0}{lim}\frac{1-cos3x}{x^2\left(\frac{3^x-1}{x}\right)}$

From (1) and (2) we get

$=\frac{9}{2log3}$