Question

For the curve represented implicitly as $3^x-2^y=1$, the value of $\underset{x\to \infty }{lim}\left(\frac{dy}{dx}\right)$ is

• A. equal to 1
• B. equal to 0
• C. equal to $lo{g}_{2}3$
• D. non existent

Answer 1

The correct option is C

equal to $lo{g}_{2}3$

$2^y=3^x-1$
$ylog2=log(3^x-1)$
$\left(log2\right)\frac{dy}{dx}=\frac{3^x.log3}{3^x-1}=\frac{log3}{1-3^{-x}}$
$\underset{x\to \infty }{Lim}\frac{dy}{dx}=\frac{log3}{log2}.\underset{x\to \infty }{lim}\frac{1}{1-3^{-x}}=lo{g}_{2}3$