## Calculus with Applications (10th Edition)

$${\text{The limit does not exist}}$$
\eqalign{ & \mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {{x^2} - 5x + 4} }}{x} \cr & {\text{Verify if the conditions of l'Hospital's rule are satisfied}}{\text{. Here}} \cr & {\text{evaluate the limit substituting }}0{\text{ for }}x{\text{ into the numerator and }} \cr & {\text{denominator}}{\text{.}} \cr & \mathop {\lim }\limits_{x \to 0} \left( {\sqrt {{x^2} - 5x + 4} } \right) = \sqrt {{{\left( 0 \right)}^2} - 5\left( 0 \right) + 4} = 2 \cr & \mathop {\lim }\limits_{x \to 0} x = 0 \cr & {\text{Since the limits do not lead to the indeterminate form }}\frac{0}{0}{\text{ or }}\frac{{ \pm \infty }}{{ \pm \infty }} \cr & {\text{we cannot apply the l'Hospital's rule}}. \cr & \cr & {\text{By l'Hospital' rule}}{\text{, the limit does not exist}} \cr}