Answer
$${\text{False}}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{x \to 0} \left[ {\frac{{{x^2} + x + 1}}{x}} \right] \cr
& {\text{Evaluating the limit by direct substitution}} \cr
& \mathop {\lim }\limits_{x \to 0} \left[ {\frac{{{x^2} + x + 1}}{x}} \right] = \mathop {\lim }\limits_{x \to 0} \frac{{{0^2} + 0 + 1}}{0} = \frac{1}{0} = \infty \cr
& {\text{The statement is false: L'Hopital's rule does not apply}} \cr
& {\text{for }}\frac{1}{0}. \cr
& {\text{False}} \cr} $$
