If the function f(x)= array|c|c| 72^x-9^x-8^x+1√2-√1+ x, & x 0 a _e2 _e3 & ,x=0 array is c…

Question

If the function

$f(x)= \begin{array}{|c|c|} \frac{{72}^{x}-{9}^{x}-{8}^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x
e 0 \ a{\log }_{e}2{\log }_{e}3 & ,x=0 \end{array}$

is continuous at $x=0$, then the value of ${a}^{2}$ is equal to

Accepted Answer

$f(x)= \begin{gathered} \frac{{72}^{x}-{9}^{x}-{8}^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}},\, x
e 0 \ a{\log }_{e}2{\log }_{e}3\, \, \, \, \, \, ,\, \, x=0 \end{gathered}$

$\because f(x)$ is continuous at $x=0$

$\begin{gathered} \Rightarrow \underset{x	o 0}{lim}\frac{{72}^{x}-{9}^{x}-{8}^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}} \ \underset{x	o 0}{lim}\frac{({9}^{x}-1)({8}^{x}-1)(\sqrt{2}+\sqrt{1+\cos x})}{\frac{(1-\cos x)}{{x}^{2}}	imes {x}^{2}} \ =(\ln 9\cdot \ln 8)(2\sqrt{2})	imes 2 \ =4\sqrt{2}	imes 2	imes 3\ln 2\cdot \ln 3 \ 24\sqrt{2}\cdot \ln 2\cdot \ln 3 \ \Rightarrow \, a=24\sqrt{2} \ \, {a}^{2}=1152 \end{gathered}$

Solutions · 1