Question

$\underset{x\to 0}{lim}{\left(\frac{2^{\sin x}+3^{\tan x}}{2}\right)}^{\csc x}$

• A. $e^{\frac{1}{2}\log 3}$
• B. $e^{\frac{1}{2}\log 6}$
• C. $e^{\frac{1}{2}\log 2}$
• D. $0$

Answer 1

The correct option is B $e^{\frac{1}{2}\log 6}$

${lim}_{x\to 0}{\left(\frac{2^{sinx}+3^{tanx}}{2}\right)}^{cosecx}$ ----------- ($1^{\infty }$ form)

$=exp{lim}_{x\to 0}cosecx\left(\frac{2^{sinx}+3^{tanx}}{2}\right)=exp({lim}_{x\to 0}\frac{2^{sinx}-1}{2sinx}+{lim}_{x\to 0}\frac{3^{tanx}-1}{2sinx})=exp\left((\frac{log2}{2}+\frac{log3}{2}){lim}_{x\to 0}\frac{1}{cosx}\right)=e^{\frac{1}{2}log6}$
Hence, option 'B' is correct.