Find the limit of ((n^2)/(3n^2+2n+1))^(1/n) as n approaches infinity

 Exercise

$\lim_{n\to\infty}\left(\sqrt[n]{\frac{n^2}{3.n^2+2.n\:+\:1}}\right)$

 

Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$

$\lim_{n\to\infty }\left(e^{\frac{1}{n}\ln\left(\frac{n^2}{3n^2+2n+1}\right)}\right)$

 

Multiplying the fraction by $\ln\left(\frac{n^2}{3n^2+2n+1}\right)$

$\lim_{n\to\infty }\left(e^{\frac{\ln\left(\frac{n^2}{3n^2+2n+1}\right)}{n}}\right)$

 

Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$

${\left(\lim_{n\to\infty }\left(e\right)\right)}^{\lim_{n\to\infty }\left(\frac{\ln\left(\frac{n^2}{3n^2+2n+1}\right)}{n}\right)}$

 

The limit of a constant is just the constant

$e^{\lim_{n\to\infty }\left(\frac{\ln\left(\frac{n^2}{3n^2+2n+1}\right)}{n}\right)}$

 

Evaluate the limit $\lim_{n\to\infty }\left(\frac{\ln\left(\frac{n^2}{3n^2+2n+1}\right)}{n}\right)$ by replacing all occurrences of $n$ by $\infty $

indeterminate

indeterminate

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