$$\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{{x^2} + 5x + 3}}{{{x^2} + x + 2}}} \right)^x} = ?$$

Solution :
$$\eqalign{ & \mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{{x^2} + 5x + 3}}{{{x^2} + x + 2}}} \right)^x} \cr & = \mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{{4x + 1}}{{{x^2} + x + 2}}} \right)^x} \cr & = \mathop {\lim }\limits_{x \to \infty } {\left[ {{{\left( {1 + \frac{{4x + 1}}{{{x^2} + x + 2}}} \right)}^{\frac{{{x^2} + x + 2}}{{4x + 1}}}}} \right]^{\frac{{\left( {4x + 1} \right)x}}{{{x^2} + x + 2}}}} \cr & = \mathop {\lim }\limits_{{e^{x \to \infty }}} \frac{{4{x^2} + x}}{{{x^2} + x + 2}} \cr & = {e^{\mathop {\lim }\limits_{x \to \infty } \frac{{4 + \frac{1}{x}}}{{1 + \frac{1}{x} + \frac{2}{{{x^2}}}}}}} \cr & = {e^4} \cr} $$