Hospital/Rational function/3x^2-5x-2 over x^3-4x^2+x+6/Example

The polynomials

$3x^{2}-5x-2{\text{ and }}x^{3}-4x^{2}+x+6$

have both a zero for ${}x=2$. It is therefore not immediately clear whether the limit

$\operatorname {lim} _{x\rightarrow 2}\,{\frac {3x^{2}-5x-2}{x^{3}-4x^{2}+x+6}}$

exists. Applying twice L'Hôpital's rule, we get the existence and

${}\operatorname {lim} _{x\rightarrow 2}\,{\frac {3x^{2}-5x-2}{x^{3}-4x^{2}+x+6}}=\operatorname {lim} _{x\rightarrow 2}\,{\frac {6x-5}{3x^{2}-8x+1}}={\frac {7}{-3}}=-{\frac {7}{3}}\,.$