Question

$li{m}_{x\to \infty }x\left[\log \right(x+a)-\log x]=$

• A. $e^2$
• B. $e^a$
• C. $a$
• D. $\frac{1}{a}$

Answer 1

The correct option is C $a$

$\underset{x\to \infty }{lim}x\left[\log \right(x+a)-\log x]$

$=\underset{x\to \infty }{lim}x\log \frac{(x+a)}{x}$

$=\underset{x\to \infty }{lim}{\left(\log \left(\frac{x+a}{x}\right)\right)}^x$

$=\underset{x\to \infty }{lim}x\log (1+\frac{a}{x})$

$=\underset{x\to \infty }{lim}\frac{x\log (1+\frac{a}{x})}{\frac{a}{x}}\frac{a}{x}$

$=a$