A) \[\log \left( \frac{2}{3} \right)\]
B) \[\frac{1}{2}\log \left( \frac{3}{2} \right)\]
C) \[\frac{1}{2}\log \left( \frac{2}{3} \right)\]
D) \[\log \,\left( \frac{3}{2} \right)\]
Correct Answer: A
Solution :
\[y=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{4}^{x}}-{{9}^{x}}}{x({{4}^{x}}+{{9}^{x}})}\], \[\left( \frac{0}{0}\text{form} \right)\] Using L-Hospital?s rule, \[y=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{4}^{x}}\log 4-{{9}^{x}}\log 9}{({{4}^{x}}+{{9}^{x}})+x({{4}^{x}}\log 4+{{9}^{x}}\log 9)}\] Þ \[y=\frac{\log 4-\log 9}{2}\] Þ \[y=\frac{\log {{\left( \frac{2}{3} \right)}^{2}}}{2}=\log \frac{2}{3}\].
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