The limit ( _ x 4 x -2 x^ 2 -4x ) evaluates to an indeterminate form of type (0/0 )
MATHEMATICS · Single correct · A Level · Mathematics · Algebra and Functions · Algebra and Functions
Problem
The limit x→4limx2−4xx−2 evaluates to an indeterminate form of type 0/0. Which of the following statements best explains why two distinct algebraic manipulations are generally required before direct substitution will produce the limit’s value?
(a) Multiplying by the conjugate removes every factor of x−4 from both numerator and denominator, so direct substitution works immediately; any further factoring would alter the limit.
(b) The numerator’s radical expression must first be rewritten so that it becomes a polynomial factor, and the denominator’s quadratic must then be factored so the common linear term can cancel.
(c) Algebraic rules forbid carrying out multiplication and factoring in the same step, so the two techniques have to be done sequentially by definition.
(d) Repeated applications of l’Hôpital’s Rule always produce the limit more efficiently than any algebraic method, so two derivative evaluations are preferred over factoring strategies.