Which of the following best explains why replacing the numerator and denominator of a…
MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Which of the following best explains why replacing the numerator and denominator of a fraction by their respective derivatives can assist in evaluating a limit that initially yields 0/0?
(a) Because differentiating numerator and denominator is equivalent to dividing out the common factor (x−a) that makes each one zero, so the new quotient avoids the indeterminate form.
(b) Because the derivatives eliminate the 0 values, guaranteeing the resulting limit is non-zero.
(c) The derivatives reveal the leading linear behavior of each function near the point, so their ratio approaches a finite value.
(d) Because differentiating numerator and denominator preserves the value of the limit for any indeterminate form, so the result must equal the original limit.