Question

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

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

Answer 1

Answer: (A)

$e^{d/b}$

${lim}_{x\to \infty }{[1+\frac{1}{a+bx}]}^{c+dx}$

Let $x=\frac{1}{y}$

As $x\to \infty ,$ then $y\to 0$

Now,${lim}_{y\to 0}{[1+\frac{1}{a+\frac{b}{y}}]}^{c+\frac{d}{y}}$

$={lim}_{y\to 0}{[1+\frac{y}{ay+b}]}^{\frac{cy+d}{y}}$

$={lim}_{y\to 0}{[1+\frac{y}{ay+b}]}^{\frac{cy+d}{y}\times \frac{y}{ay+b}\times \frac{cy+d}{y}}$

$={\left[{lim}_{y\to 0}{(1+\frac{y}{ay+b})}^{\frac{ay+b}{y}}\right]}^{\frac{cy+d}{ay+b}}$

$=e^{{lim}_{y\to 0}\frac{cy+d}{ay+b}}$

$=e^{\frac{d}{b}}$