If limn→∞ n⋅3nn(x−2)n+n⋅3n+1−3n=13, where n∈N, then the number of integer (s) in the range of x, is

A
3
B
4
C
5
D
infinite

Solution:

We have,

$\begin{array}{l} lim_{n \to \infty} \frac{n \cdot 3^{n}}{n (x - 2)^{n} + n \cdot 3^{n + 1} - 3^{n}} = \frac{1}{3} \\ \Rightarrow lim_{n \to \infty} \frac{1}{{(\frac{x - 2}{3})}^{n} + 3 - \frac{1}{n}} = \frac{1}{3} \\ \Rightarrow - 1 < \frac{x - 2}{3} < 1 \Rightarrow - 3 < x - 2 < 1 \Rightarrow - 1 < x < 5 \end{array}$

Hence, the possible integer values of $x$ are $0, 1, 2, 3$ and $4 .$

If $lim_{n \to \infty} \frac{n \cdot 3^{n}}{n (x - 2)^{n} + n \cdot 3^{n + 1} - 3^{n}} = \frac{1}{3},$ where $n \in N,$ then the number of integer $(s)$ in the range of $x,$ is

Solution:

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