For real parameters with and

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

For real parameters

a>0

with

a\ne 1

and

b\ne 0

, define

Q_{a,b}(x)=\frac{a^{\,x}-1}{\ln(1+bx)}\quad (x\ne 0).

Investigate the behavior of

Q_{a,b}(x)

x\to 0

by answering the questions below. (1) Determine the numerical limits of

a^{\,x}-1

and

\ln(1+bx)

individually as

x\to 0

. Hence identify all pairs

(a,b)

for which the quotient

Q_{a,b}(x)

approaches an indeterminate ratio of type

0/0

x=0

. Provide clear reasoning. (2) For parameter pairs identified in part (a), verify that both the numerator and the denominator are differentiable on an open interval containing

x=0

(except possibly at

x=0

itself). Cite relevant derivative formulas to support your answer. (3) Construct the first-derivative quotient that would naturally follow from the analysis in part (b). Present your result in simplified analytic form, but do not evaluate its limit. (4) Discuss whether further successive replacements of numerator and denominator by their derivatives could ever be necessary for the limit of

Q_{a,b}(x)

x=0

, and justify your answer in terms of the algebraic structure of the functions involved.

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