Question

Evaluate $\underset{y\to 0}{lim}\left[\underset{x\to \infty }{lim}\frac{exp\left[x\ell n[1+\frac{ay}{x}]\right]-exp\left[x\ell n[1+\frac{by}{x}]\right]}{y}\right]$

• A. $a-b$
• B. $a+b$
• C. $b-a$
• D. $-a-b$

Answer 1

The correct option is A $a-b$

$\underset{y\to 0}{lim}\left[\underset{x\to \infty }{lim}\frac{e^{x\ln (1+\frac{a}{y})}-e^{x\ln (1+\frac{b}{y})}}{y}\right]$

Rewriting the above equation
$\Rightarrow \underset{y\to 0}{lim}\left[\underset{x\to \infty }{lim}\frac{e^{\frac{\ln (1+\frac{ay}{x})}{\frac{ay}{x}\times \frac{1}{ay}}}-e^{\frac{\ln (1+\frac{by}{x})}{\frac{by}{x}\times \frac{1}{by}}}}{y}\right]$

$=\underset{y\to 0}{lim}\left\{\frac{e^{ay}-e^{by}}{y}\right\}=\underset{y\to 0}{lim}[a\frac{(e^{ay}-1)}{ay}-b\frac{(e^{by}-1)}{by}]\Rightarrow a-b$