find-the-limit-as-n-gt-lim-2-n-x-n-

Question Number 81970 by arkanmath7@gmail.com last updated on 17/Feb/20

f i n d t h e l i m i t a s n - > \infty

l i m {(2 -^{n} \sqrt{x})}^{n}

Commented by msup trace by abdo last updated on 17/Feb/20

l e t f (x) = {(2 -^{n} \sqrt{x})}^{n} \Rightarrow

f (x) = e^{n l n (2 -^{n} \sqrt{x})}

= e^{n l n (2) + l n (1 - \frac{1}{2} e^{\frac{1}{n} l n (x)})}

= e^{n l n (2)} \times e^{l n (1 - \frac{1}{2} e^{\frac{l n x}{n}})} \to + \infty

b e c a u s e {l i m}_{n \to + \infty} e^{n l n (2)} = + \infty