Question

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$\Large{\underline{\underline{\mathfrak{\pink{bf{Question}}}}}}$
★Find the value of ......
$\displaystyle \lim_{x \to \infty } {x}^{\dfrac{1}{x}}$
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Answer 1

Step-by-step explanation:

you can understand by seeing the pic

Answer 2

The value of $\lim _{x\to \infty }\left(x^{\frac{1}{x}}\right)$ is as follows

Given,

$\lim _{x\to \infty }\left(x^{\frac{1}{x}}\right)$

Applying exponential rule,

$a^x=e^{\ln \left(a^x\right)}=e^{x\cdot \ln \left(a\right)}$

we get,

$x^{\frac{1}{x}}=e^{\frac{1}{x}\ln \left(x\right)}$

$=\lim _{x\to \infty \:}\left(e^{\frac{1}{x}\ln \left(x\right)}\right)$

= 1

Therefore the value of $\lim _{x\to \infty }\left(x^{\frac{1}{x}}\right)$ is 1.