Question

$\displaystyle \lim_{x \to 2}$ $\frac{x^{100}-2^{100}}{x^{77}-2^{77}}$ is equal to

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

Answer 1

Answer: (D)

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

Answer 2

$\displaystyle\lim _{x \rightarrow 2} \frac{x^{100}-2^{100}}{x^{77}-2^{77}}$
$=\displaystyle\lim _{x \rightarrow 2} \frac{x^{100}-2^{100}}{x-2} \times \frac{x-2}{x^{77}-2^{77}}$
$\left(\because \,\displaystyle\lim _{x \rightarrow a} \frac{x^{m} \quad a^{m}}{x-a}=m a^{m-1}\right)$
$=\displaystyle\lim _{x \rightarrow 2}\left(\frac{x^{100}-2^{100}}{x-2}\right) \times \frac{1}{\displaystyle\lim _{x \rightarrow 2}\left(\frac{x^{77}-2^{77}}{x-2}\right)}$
$=100(2)^{99} \times \frac{1}{77(2)^{76}}$
$=\frac{100}{77}(2)^{23}$