Question
Download Solution PDF\(\mathop {\lim }\limits_{n \to \infty } \frac{{{2^{n + 1}} + {3^{n + 1}}}}{{{2^n} + {3^n}}}\) किसके बराबर है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDF\(\mathop {\lim }\limits_{n \to \infty } \frac{{{2^{n + 1}} + {3^{n + 1}}}}{{{2^n} + {3^n}}} \)
इसे इस प्रकार लिखा जा सकता है:
\(= \mathop {\lim }\limits_{n \to \infty } \frac{{{2^n}2 + {3^n}3}}{{{2^n} + {3^n}}}\)
3n आम लेते हुए, हम लिख सकते हैं:
\( = \mathop {\lim }\limits_{n \to \infty } \frac{{{3^n}\left[ {2.{{\left( {\frac{2}{3}} \right)}^n} + 3} \right]}}{{{3^n}\left[ {{{\left( {\frac{2}{3}} \right)}^n} + 1} \right]}}\)
\( = \mathop {\lim }\limits_{n \to \infty } \frac{{2.{{\left( {\frac{2}{3}} \right)}^n} + 3}}{{\left[ {{{\left( {\frac{2}{3}} \right)}^n} + 1} \right]}}\)
यहाँ \(\frac 2 3 < 1\)
इसलिए, \(\left[ \frac {2}{3}\right]^{\infty} = 0\)
\(= \frac{{0 + 3}}{{0 + 1}} = 3\)
Last updated on May 26, 2025
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