Question

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

• A. $\frac{100}{77}$
• B. $\frac{100}{77}\left(2^{22}\right)$
• C. $\frac{100}{77}\left(2^{21}\right)$
• D. $\frac{100}{77}\left(2^{23}\right)$
• E. $\frac{100}{77}\left(2^{24}\right)$

Answer 1

Answer: (D)

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