Question

If a, b, c, d are positive, then $\lim_{x \rightarrow \infty}\left( 1 + \frac{1}{a + bx} \right)^{c + dx}$

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

Answer 1

Answer: (A)

$e^{d/b}$

Answer 2

$\lim_{x \rightarrow \infty}\left( 1 + \frac{1}{a + bx} \right)^{c + dx} = \lim_{x \rightarrow \infty}\left\{ \left( 1 + \frac{1}{a + bx} \right)^{a + bx} \right\}^{\frac{c + dx}{a + bx}}$

= $e^{d/b}\left\{ \because\lim_{x \rightarrow \infty}\left( 1 + \frac{1}{a + bx} \right)^{a + bx} = e \right.\$ and $\left. \ \lim_{x \rightarrow \infty}\frac{c + dx}{a + bx} = \frac{d}{b} \right\}$

Alternative method : $e^{\lim_{x \rightarrow \infty}\left( \frac{1}{a + bx} \right)\left( \frac{c + dx}{1} \right)} = e^{d/b}$.