real analysis - Evaluate $lim_{n to infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{frac{1}{n}}$

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$ Here [.] denotes the greatest integer function.

My Try : I know how to solve this kind of problem :$\lim_{n \to \infty} ((a)^n +(b)^n)^{\frac{1}{n}}$ where $a, b \geq 0$. But here I can not find $([(1+0.0001)^{10000}])$?

Can anyone please help me out?

Thank You.

Answer

Since for all $n\in \mathbb{N}$

$$2\le\left(1+\frac{1}{n}\right)^n \le e < 3$$

we have that

$$\left((15)^n +\left[\left(1+\frac1{10000}\right)^{10000}\right]^n\right)^{\frac{1}{n}}= \left((15)^n +2^n\right)^{\frac{1}{n}}=15 \left(1 +(2/15)^n\right)^{\frac{1}{n}}\to 15$$