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Question Number 162790 by LEKOUMA last updated on 01/Jan/22

$$Calculate$$$$\underset{n\to \mathrm{\infty }}{\lim }{(1+2^n+3^n)}^{\frac{1}{n}}$$

Answered by mindispower last updated on 01/Jan/22

$$=\underset{n\to \mathrm{\infty }}{\lim }{e}^{\frac{ln\left(3^n\right)+ln(\frac{1}{3^n}+{\left(\frac{2}{3}\right)}^n+1)}{n}}$$$$=\underset{n\to \mathrm{\infty }}{\lim }{e}^{ln\left(3\right)}.e^{ln(1+\frac{1}{3^n}+{\left(\frac{2}{3}\right)}^n)}=3$$$$\frac{1}{3^n}\to 0,{\left(\frac{2}{3}\right)}^n\to 0$$

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