How-to-solve-this-limit-lim-x-7x-2-x-x-

Question Number 66084 by F_Nongue last updated on 09/Aug/19

H o w t o s o l v e t h i s l i m i t ?

\underset{x \to \infty}{l i m} {(7 x + \frac{2}{x})}^{x}

Commented by mathmax by abdo last updated on 09/Aug/19

l e t A (x) = {(7 x + \frac{2}{x})}^{x} \Rightarrow A (x) = {(7 x)}^{x} {(1 + \frac{2}{7 x^{2}})}^{x}

= {(7 x)}^{x} e^{x l n (1 + \frac{2}{7 x^{2}})} b u t l n (1 + \frac{2}{7 x^{2}}) \sim \frac{2}{7 x^{2}} \Rightarrow x l n (1 + \frac{2}{7 x^{2}}) \sim \frac{2}{7 x} \Rightarrow

A (x) \sim {(7 x)}^{x} \times \frac{2}{7 x} = 2 {(7 x)}^{x - 1} = 2 e^{(x - 1) l n (7 x)} \to + \infty (x \to + \infty) \Rightarrow

{l i m}_{x \to + \infty} {(7 x + \frac{2}{x})}^{x} = + \infty

Answered by MJS last updated on 09/Aug/19

{(7 x + \frac{2}{x})}^{x} = {(\frac{7 x^{2} + 2}{x})}^{x} = \frac{{(7 x^{2} + 2)}^{x}}{x^{x}}

\lim_{x \to \infty} \frac{{(7 x^{2} + 2)}^{x}}{x^{x}} > \lim_{x \to \infty} \frac{{(7 x^{2})}^{x}}{x^{x}} = {\underset{x \to \infty}{\lim 7}}^{x} x^{x} = + \infty

Commented by Prithwish sen last updated on 09/Aug/19

\frac{2}{x} \to 0 as x \to \infty ∴ {(7 x + \frac{2}{x})}^{x} \to 7^{x} x^{x}