Question

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

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

Answer 1

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

Form of the limit is $1^{\infty }$
$\therefore \underset{x\to \infty }{lim}{(1+\frac{1}{a+bx})}^{c+dx}=\underset{x\to \infty }{lim}{(1+\frac{1}{a+bx})}^{(a+bx)\cdot \frac{c+dx}{a+bx}}$
$=e^{\underset{x\to \infty }{lim}\frac{c+dx}{a+bx}}=e^{\underset{x\to \infty }{lim}\frac{c/x+d}{a/x+b}}=e^{d/b}$