Define the function q(x)= cases x x , & x<0

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

Problem

Define the function

q(x)=\begin{cases} \dfrac{\sin x}{x}, & x<0,\\ cx^2+dx+e, & 0\le x<\pi,\\ 2\cos x, & x\ge \pi, \end{cases}

where

c

d

, and

e

are integers. The goal is to make

q

continuous on the entire real line while additionally requiring

c

to be the smallest positive integer for which such continuity is possible. (1) Part 1: (a) Derive the equation relating

d

and

e

that arises from continuity of

q

x=0

. Show all limit calculations and justify any indeterminate forms you evaluate. (2) Part 2: (b) Derive the equation relating

c

d

, and

e

that arises from continuity of

q

x=\pi

. (3) Part 3: (c) Solve the simultaneous equations obtained in parts (a) and (b) to express

d

and

e

in terms of

c

. (4) Part 4: (d) Determine the least positive integer value of

c

that yields integer values of

d

and

e

satisfying both continuity conditions. State the corresponding ordered triple

(c,d,e)

. (5) Part 5: (e) Provide a rigorous argument confirming that, with the triple found in part (d),

q

is continuous at both

x=0

and

x=\pi

, and hence on the entire real line.