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12. limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS12. limits1 Mark
MCQ
$\mathop {\lim }\limits_{n \to \infty } {({4^n} + {5^n})^{1/n}}$ is equal to
  • A
    $4$
  • ✓
    $5$
  • C
    $e$
  • D
    None of these
✓

Answer

Correct option: B.
$5$
b
(b) Given limit $ = \mathop {\lim }\limits_{n \to \infty } \,{({4^n} + {5^n})^{1/n}}$

$ = \mathop {\lim }\limits_{n \to \infty } \,5\,{\left[ {{{\left\{ {1 + {{\left( {\frac{4}{5}} \right)}^n}} \right\}}^{{{(5/4)}^n}}}} \right]^{(1/n)\,.\,{{(4/5)}^n}}} = 5\,.\,{e^0} = 5$.

$\left( {\because \,\,{{\left( {\frac{4}{5}} \right)}^n} \to 0\,{\text{as }}\,n \to \infty } \right)$

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