Question

The value of $\displaystyle\lim _{x \rightarrow 3}\left(\log _{a} \frac{x-3}{\sqrt{x+6}-3}\right)$ is

• A. $\log _{a} 6$
• B. $\log _{a} 3$
• C. $\log _{a} 2$
• D. None of these

Answer 1

Answer: (A)

$\displaystyle\lim _{x \rightarrow 3}\left[\log _{a} \frac{x-3}{\sqrt{x+6}-3}\right]$
$=\displaystyle\lim _{x \rightarrow 3}\left[\log _{a} \frac{(x-3)(\sqrt{x+6}+3)}{(x-3)}\right]$
$=\displaystyle\lim _{x \rightarrow 3} \log _{a}(\sqrt{x+6}+3)=\log _{a} 6$
$\displaystyle\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}$
where $n \in Q$, the set of rational numbers.