Question

The value of $\underset{x\to \infty }{lim}(3^x+4^x{)}^{1/x}$ is

• A. $\frac{4}{3}$
• B. $4$
• C. $4e$
• D. $4+e$

Answer 1

The correct option is B $4$

$L=\underset{x\to \infty }{lim}(3^x+4^x{)}^{1/x}=\underset{x\to \infty }{lim}4{[{\left(\frac{3}{4}\right)}^x+1]}^{1/x}$
Let $y={\left(\frac{3}{4}\right)}^x$
As $x\to \infty ,y\to 0$
$L=4\underset{x\to \infty }{lim}(1+y{)}^{1/x}$
$L=4\underset{x\to \infty }{lim}{\left[\right(1+y{)}^{1/y}]}^{y/x}$
As $x\to \infty ,y\to 0\Rightarrow \frac{y}{x}\to 0$
$\therefore L=4\times e^0=4$