prove-that-lim-x-x-1-x-1-

Question Number 75694 by malwaan last updated on 15/Dec/19

p r o v e t h a t

\underset{x \to \infty}{l i m} x^{\frac{1}{x}} = 1

Answered by vishalbhardwaj last updated on 15/Dec/19

(a s x \to \infty then \frac{1}{x} \to 0)

\Rightarrow l i {\underset{x \to \infty}{m x}}^{l i \underset{x \to \infty}{m} \frac{1}{x}} = l i \underset{x \to \infty}{m} {(x)}^{0} = 1

Commented by malwaan last updated on 15/Dec/19

b u t \underset{x \to \infty}{l i m} x^{0} = \infty^{0} u n d e f i n d

Commented by vishalbhardwaj last updated on 15/Dec/19

whatever the value if x is there, but

the whole power is zero, that why

its value will be zero in the sense

Commented by MJS last updated on 15/Dec/19

you cannot split it like this

\lim_{x \to \infty} x^{\frac{1}{x}} \neq {(\lim_{x \to \infty} x)}^{(\lim_{x \to \infty} \frac{1}{x})}

because \lim_{x \to \infty} x = \infty \notin R

and r^{0} = 1 only for r \in R ∖ {0}

Answered by MJS last updated on 15/Dec/19

\lim_{x \to \infty} x^{\frac{1}{x}} = {\underset{x \to \infty}{\lim e}}^{\frac{\ln x}{x}} = e^{\lim_{x \to \infty} \frac{\ln x}{x}}

\lim_{x \to \infty} \frac{\ln x}{x} = \lim_{x \to \infty} \frac{\frac{d}{d x} [\ln x]}{\frac{d}{d x} [x]} = \lim_{x \to \infty} \frac{1}{x} = 0

\Rightarrow e^{\lim_{x \to \infty} \frac{\ln x}{x}} = 1 \Rightarrow {\underset{x \to \infty}{\lim e}}^{\frac{\ln x}{x}} = 1 \Rightarrow \lim_{x \to \infty} x^{\frac{1}{x}} = 1

Commented by vishalbhardwaj last updated on 15/Dec/19

sir, Is my assumption and explantion wrong ? ?

Commented by malwaan last updated on 16/Dec/19

t h a n k y o u s o m u c h

Answered by $@ty@m123 last updated on 15/Dec/19

S o l u t i o n :

L e t y = \lim_{x \to \infty} x^{\frac{1}{x}}

\ln y = \lim_{x \to \infty} \frac{1}{x} \ln x

\Rightarrow \ln y = 0

\Rightarrow y = 1