_ n ( 3^n + 4^n )^ 1 n = - Vidyadip

Question

$\mathop {\lim }\limits_{n \to \infty } {({3^n} + {4^n})^{\frac{1}{n}}} = $

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Answer

b
(b) $\mathop {{\rm{lim}}}\limits_{n \to \infty } \,{({3^n} + {4^n})^{\frac{1}{n}}}$

$ = \mathop {{\rm{lim}}}\limits_{n \to \infty } \,{({4^n})^{\frac{1}{n}}}{\left[ {\frac{{{3^n}}}{{{4^n}}} + 1} \right]^{\frac{1}{n}}}$

$ = \mathop {{\rm{lim}}}\limits_{n \to \infty } 4\,{\left[ {1 + \frac{1}{{{{\left( {\frac{4}{3}} \right)}^n}}}} \right]^{1/n}}$

$ = 4\mathop {{\rm{lim}}}\limits_{n \to \infty } \,{\left[ {1 + \frac{1}{{{{\left( {\frac{4}{3}} \right)}^n}}}} \right]^{1/n}}$

$ = 4{\left[ {1 + \frac{1}{\infty }} \right]^0} = 4 \times {(1)^0}$ $ = 4 \times 1 = 4$.

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