Evaluate: \lim_{x arrow+infinity} (((x^2+1)/(x^2-2))^{(x^2)})

Expression: $\lim_{x \rightarrow +\infty} \left({\left( \frac{ {x}^{2}+1 }{ {x}^{2}-2 } \right)}^{\left( {x}^{2} \right)}\right)$

Substitute $t$ for ${x}^{2}-2$

$\lim_{t \rightarrow +\infty} \left({\left( \frac{ t+2+1 }{ t } \right)}^{t+2}\right)$

$\lim_{t \rightarrow +\infty} \left({\left( \frac{ t+3 }{ t } \right)}^{t+2}\right)$

Separate the fraction into $2$ fractions

$\lim_{t \rightarrow +\infty} \left({\left( \frac{ t }{ t }+\frac{ 3 }{ t } \right)}^{t+2}\right)$

Any expression divided by itself equals $1$

$\lim_{t \rightarrow +\infty} \left({\left( 1+\frac{ 3 }{ t } \right)}^{t+2}\right)$

Use $a=b \times \frac{ a }{ b }$ to transform the expression

$\lim_{t \rightarrow +\infty} \left({\left( 1+\frac{ 3 }{ t } \right)}^{t \times \frac{ t+2 }{ t }}\right)$

Use ${a}^{mn}={\left( {a}^{m} \right)}^{n}$ to transform the expression

$\lim_{t \rightarrow +\infty} \left({\left( {\left( 1+\frac{ 3 }{ t } \right)}^{t} \right)}^{\frac{ t+2 }{ t }}\right)$

Use $\lim_{x \rightarrow c} \left({f\left( x \right)}^{g\left( x \right)}\right)={\left( \lim_{x \rightarrow c} \left(f\left( x \right)\right) \right)}^{\lim_{x \rightarrow c} \left(g\left( x \right)\right)}$ to transform the expression

${\left( \lim_{t \rightarrow +\infty} \left({\left( 1+\frac{ 3 }{ t } \right)}^{t}\right) \right)}^{\lim_{t \rightarrow +\infty} \left(\frac{ t+2 }{ t }\right)}$

Evaluate the limit

${\left( {e}^{3} \right)}^{\lim_{t \rightarrow +\infty} \left(\frac{ t+2 }{ t }\right)}$

Evaluate the limit

${\left( {e}^{3} \right)}^{1}$

Simplify the expression by multiplying exponents

\begin{align*}&{e}^{3} \\&\approx20.08554\end{align*}

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