Answer
= $1$
Work Step by Step
$f(x)$ = $\mathop {\lim }\limits_{x \to 0^{+} }$ $(1+\frac{3}{x})^{x}$
$\ln{f(x)}$ = $\mathop {\lim }\limits_{x \to 0^{+} }$ $\ln(1+\frac{3}{x})^{x}$
$\ln{f(x)}$ = $\mathop {\lim }\limits_{x \to 0^{+} }$ $x(\ln(1+3x^{-1}))$
$\ln{f(x)}$ = $\mathop {\lim }\limits_{x \to 0^{+} }$ $\frac{\ln(1+3x^{-1})}{x^{-1}}$
$\ln{f(x)}$ = $\mathop {\lim }\limits_{x \to 0^{+} }$ $\frac{\frac{1}{1+3x^{-1}}(-3x^{-2})}{-x^{-2}}$
$\ln{f(x)}$ = $\mathop {\lim }\limits_{x \to 0^{+} }$ $\frac{3}{1+3x^{-1}}$
$\ln{f(x)}$ = $\mathop {\lim }\limits_{x \to 0^{+} }$ $\frac{3x}{x+3}$
$\ln{f(x)}$ = $0$
${f(x)}$ = $e^{0}$
${f(x)}$ = $1$
