A) \[\frac{100}{77}\]
B) \[\frac{100}{77}({{2}^{22}})\]
C) \[\frac{100}{77}({{2}^{21}})\]
D) \[\frac{100}{77}({{2}^{23}})\]
E) \[\frac{100}{77}({{2}^{24}})\]
Correct Answer: D
Solution :
\[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{100}}-{{2}^{100}}}{{{x}^{77}}-{{2}^{27}}}\] \[=\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{100}}-{{2}^{100}}}{x-2}\times \frac{x-2}{{{x}^{77}}-{{2}^{77}}}\] \[\left( \because \underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{m}}-{{a}^{m}}}{x-a}=m{{a}^{m-1}} \right)\] \[=\underset{x\to 2}{\mathop{\lim }}\,\left( \frac{{{x}^{100}}-{{2}^{100}}}{x-2} \right)\times \frac{1}{\underset{x\to 2}{\mathop{\lim }}\,\left( \frac{{{x}^{77}}-{{2}^{77}}}{x-2} \right)}\] \[=100{{(2)}^{99}}\times \frac{1}{77{{(2)}^{76}}}\] \[=\frac{100}{77}{{(2)}^{23}}\]
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