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

Answer: (D)

$\frac{100}{77}({{2}^{23}})$

Answer 2

$\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{100}}-{{2}^{100}}}{{{x}^{77}}-{{2}^{27}}}$
$=\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{100}}-{{2}^{100}}}{x-2}\times \frac{x-2}{{{x}^{77}}-{{2}^{77}}}$ $\left( \because \underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{m}}-{{a}^{m}}}{x-a}=m{{a}^{m-1}} \right)$
$=\underset{x\to 2}{\mathop{\lim }}\,\left( \frac{{{x}^{100}}-{{2}^{100}}}{x-2} \right)\times \frac{1}{\underset{x\to 2}{\mathop{\lim }}\,\left( \frac{{{x}^{77}}-{{2}^{77}}}{x-2} \right)}$
$=100{{(2)}^{99}}\times \frac{1}{77{{(2)}^{76}}}$
$=\frac{100}{77}{{(2)}^{23}}$