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Question Number 35715 by math1967 last updated on 22/May/18

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

Commented by abdo mathsup 649 cc last updated on 22/May/18

$$\text{Missing left or extra right}$$$$=3{lim}_{n\to +\mathrm{\infty }}\frac{1+{\left(\frac{2}{3}\right)}^{n+1}}{1+{\left(\frac{2}{3}\right)}^n}=3dueto$$$$lim{lim}_{n\to +\mathrm{\infty }}{\left(\frac{2}{3}\right)}^n=0$$

Answered by ajfour last updated on 22/May/18

$$=\underset{n\to \mathrm{\infty }}{\lim }\frac{3(3^n+\frac{2}{3}\times 2^n)}{3^n+2^n}$$$$=\underset{n\to \mathrm{\infty }}{\lim 3}(1-\frac{2^n}{3^n+2^n})$$$$=\underset{n\to \mathrm{\infty }}{\lim 3}(1-\frac{1}{{\left(\frac{3}{2}\right)}^n+1})=3.$$

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