Let be a real constant

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

a

be a real constant. Define the functions •

h(x)=a\,x

, •

g(y)=\sin y

, and •

f(t)=\ln(1+t)

, whose domain is

t>-1

. Consider the composition

L(x)=f\bigl(g\bigl(h(x)\bigr)\bigr)=\ln\bigl(1+\sin(ax)\bigr)

. The limit

\displaystyle \lim_{x\to 0}L(x)

exists and is finite for some, but not necessarily all, real values of

a

. (a) Determine all real values of

a

for which the limit exists and is finite. Provide a clear explanation that addresses both existence and finiteness. (b) For each value of

a

you identified in part (a), evaluate

\displaystyle \lim_{x\to 0}L(x)

, showing all relevant steps and justifications.