\(\lim_{x \to \infty} (\frac{x+7}{x+2})^{x+5}\) equal to
\[ \mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{x + 7}}{{x + 2}}} \right)^{x + 5}} = \mathop {\lim }\limits_{x \to \infty } {\left( {{{\left( {1 + \frac{5}{{x + 2}}} \right)}^{x + 2}}} \right)^{\frac{{x + 5}}{{x + 2}}}} \] \[ = \mathop {\lim }\limits_{x \to \infty } {\left( {{e^5}} \right)^{\frac{{x + 5}}{{x + 2}}}} = {e^5} \]
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