Evaluate: \lim_{x arrow+infinity} ((sqrt(x^3-4x+5)+10)/(2x^2-6))

Expression: $\lim_{x \rightarrow +\infty} \left(\frac{ \sqrt{ {x}^{3}-4x+5 }+10 }{ 2{x}^{2}-6 }\right)$

Evaluate the limits of numerator and denominator separately

$\begin{array} { l }\lim_{x \rightarrow +\infty} \left(\sqrt{ {x}^{3}-4x+5 }+10\right),\\\lim_{x \rightarrow +\infty} \left(2{x}^{2}-6\right)\end{array}$

Evaluate the limit

$\begin{array} { l }+\infty,\\\lim_{x \rightarrow +\infty} \left(2{x}^{2}-6\right)\end{array}$

Evaluate the limit

$\begin{array} { l }+\infty,\\+\infty\end{array}$

Since the expression $\frac{ +\infty }{ +\infty }$ is an indeterminate form, try transforming the expression

$\lim_{x \rightarrow +\infty} \left(\frac{ \sqrt{ {x}^{3}-4x+5 }+10 }{ 2{x}^{2}-6 }\right)$

Factor out ${x}^{2}$ from the expression

$\lim_{x \rightarrow +\infty} \left(\frac{ {x}^{2} \times \left( \sqrt{ \frac{ 1 }{ x }-\frac{ 4 }{ {x}^{3} }+\frac{ 5 }{ {x}^{4} } }+\frac{ 10 }{ {x}^{2} } \right) }{ 2{x}^{2}-6 }\right)$

Factor out ${x}^{2}$ from the expression

$\lim_{x \rightarrow +\infty} \left(\frac{ {x}^{2} \times \left( \sqrt{ \frac{ 1 }{ x }-\frac{ 4 }{ {x}^{3} }+\frac{ 5 }{ {x}^{4} } }+\frac{ 10 }{ {x}^{2} } \right) }{ {x}^{2} \times \left( 2-\frac{ 6 }{ {x}^{2} } \right) }\right)$

Cancel out the common factor ${x}^{2}$

$\lim_{x \rightarrow +\infty} \left(\frac{ \sqrt{ \frac{ 1 }{ x }-\frac{ 4 }{ {x}^{3} }+\frac{ 5 }{ {x}^{4} } }+\frac{ 10 }{ {x}^{2} } }{ 2-\frac{ 6 }{ {x}^{2} } }\right)$

Evaluate the limit

$\frac{ \sqrt{ 0-4 \times 0+5 \times 0 }+10 \times 0 }{ 2-6 \times 0 }$

Simplify the expression

$0$