A) 3
B) 4
C) \[\infty \]
D) e
Correct Answer: B
Solution :
\[\underset{n\to \infty }{\mathop{\text{lim}}}\,\,{{({{3}^{n}}+{{4}^{n}})}^{\frac{1}{n}}}\] \[=\underset{n\to \infty }{\mathop{\text{lim}}}\,\,{{({{4}^{n}})}^{\frac{1}{n}}}{{\left[ \frac{{{3}^{n}}}{{{4}^{n}}}+1 \right]}^{\frac{1}{n}}}\] \[=\underset{n\to \infty }{\mathop{\text{lim}}}\,4\,{{\left[ 1+\frac{1}{{{\left( \frac{4}{3} \right)}^{n}}} \right]}^{1/n}}\]\[=4\underset{n\to \infty }{\mathop{\text{lim}}}\,\,{{\left[ 1+\frac{1}{{{\left( \frac{4}{3} \right)}^{n}}} \right]}^{1/n}}\] \[=4{{\left[ 1+\frac{1}{\infty } \right]}^{0}}=4\times {{(1)}^{0}}\] \[=4\times 1=4\].
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