Question

$\underset{x\to 0}{lim}\left[\begin{array}{c}\frac{3^x+3^{-x}-2}{x^2}\end{array}\right]$ is equal to

• A. 2 log 3
• B. $-(log3{)}^2$
• C. -2 log 3
• D. $(log3{)}^2$

Answer 1

The correct option is D $(log3{)}^2$

$\underset{x\to 0}{lim}\left[\frac{3^x+36-x-2}{x^2}\right]$
$=\underset{x\to 0}{lim}\frac{\left[\begin{array}{c}1+xlog3+\frac{x^2}{2!}(log3{)}^2+...\\ +1-xlog3+\frac{x^2}{2!}(log3{)}^2-...-2\end{array}\right]}{x^2}$
$=\underset{x\to 0}{lim}(log3{)}^2+\{\text{terms containing power greater than 2}\}$
$=(log3{)}^2$