Question

If a, b, c, d are positive, then $\underset{x\to \infty }{lim}{(1+\frac{1}{a+bx})}^{c+dx}$ is equal to

• A. $e^{d/b}$
• B. $e^{d/a}$
• C. $e^{(c+d)/a+b}$
• D. $e$

Answer 1

The correct option is A $e^{d/b}$

$\underset{x\to \infty }{lim}{(1+\frac{1}{a+bx})}^{c+dx}$

$=\underset{x\to \infty }{lim}{(1+\frac{1}{a+bx})}^{(a+bx)\frac{c+dx}{a+bx}}$

$=\underset{x\to \infty }{lim}{(1+\frac{1}{a+bx})}^{(a+bx){lim}_x\to \infty \left(\frac{c/x+d}{a/x+b}\right)}$..............$\underset{x\to \infty }{lim}{(1+\frac{1}{x})}^x=e$

$=e^{d/b}$

Hence, option ${}^{\prime }{A}^{\prime }$ is correct.