Evaluate: \lim_{x arrow 0} ((2x^2-3x+4)/(x)+(5x-4)/(x))

Expression: $\lim_{x \rightarrow 0} \left(\frac{ 2{x}^{2}-3x+4 }{ x }+\frac{ 5x-4 }{ x }\right)$

Write all numerators above the common denominator

$\lim_{x \rightarrow 0} \left(\frac{ 2{x}^{2}-3x+4+5x-4 }{ x }\right)$

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

$\lim_{x \rightarrow 0} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 2{x}^{2}-3x+4+5x-4 \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ 4x+2 }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ 4x+2 }{ 1 }\right)$

Any expression divided by $1$ remains the same

$\lim_{x \rightarrow 0} \left(4x+2\right)$

Evaluate the limit

$4 \times 0+2$

Simplify the expression

$2$