L’Hospital’s rule in Calculus gives us a way to evaluate the limit of an indeterminate quotient. An indeterminate quotient or fraction is one where both the numerator and denominator are either zero OR infinity.
What is L’Hospital’s Rule? It is a rule for finding a limit of a quotient that essentially states that it is OK to take the derivative of BOTH the numerator and denominator until you get a numerical answer for your limit.
The following quotient of ( tangent minus x ) divided by x cubed as x approaches 0 requires 3 applications of L’Hospital’s rule before a determinate limit is reached. Also note the trig Identity 1 plus tangent squared equals to secant squared was used along with the limit law that says you can split the limit with respect to the operation of multiplication.
The chain rule was also used in going from the third to the fourth fraction.