Calculate: \lim_{x arrow+infinity} ((x^2+2)/(x-3))

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

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow +\infty} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{2}+2 \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x-3 \right) }\right)$

Find the derivative

$\lim_{x \rightarrow +\infty} \left(\frac{ 2x }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x-3 \right) }\right)$

Find the derivative

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

Any expression divided by $1$ remains the same

$\lim_{x \rightarrow +\infty} \left(2x\right)$

The limit at $+\infty$ of a polynomial whose leading coefficient is positive equals $+\infty$

$+\infty$