Question

$ \underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{100}}-{{2}^{100}}}{{{x}^{77}}-{{2}^{77}}} $ is equal to

• A. $ \frac{100}{77} $
• B. $ \frac{100}{77}({{2}^{22}}) $
• C. $ \frac{100}{77}({{2}^{21}}) $
• D. $ \frac{100}{77}({{2}^{23}}) $

Answer 1

The Correct Option is D

$ \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}} $