Question

$\underset{x\to y}{lim}\frac{X^Y-Y^X}{X^X-Y^Y}$ is equal to

• A. $\frac{1+\log Y}{\log Y}$
• B. $\frac{1-\log Y}{1+\log Y}$
• C. $\frac{1-\log Y}{\log \left(eY\right)}$
• D. $\log \left(eX\right)$

Answer 1

The correct option is B $\frac{1-\log Y}{1+\log Y}$

${lim}_{x\to y}\frac{X^Y-Y^X}{X^X-Y^Y}$

Apply L hospital rule

${lim}_{x\to y}\frac{y{x}^{y-1}-y^x\log y}{x^x(1+\log x)}$
$=\frac{y{y}^{y-1}-y^y\log y}{y^y(1+\log y)}$
$=\frac{y^y(1-\log y)}{y^y(1+\log y)}$
$=\frac{1-\log y}{1+\log y}$