A parameter-dependent function is defined by s(x)= cases - x+2 x-c , & x<c

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

Problem

A parameter-dependent function

s

is defined by

s(x)=\begin{cases} -\dfrac{x+2}{x-c}, & x<c,\\ e^{x-c}, & x\ge c, \end{cases}

where

c

is a real parameter. The location of the breakpoint

x=c

is allowed to slide along the real line. You will explore how different choices of

c

affect the limit and continuity properties of

s

x=c

. (1) Part 1: (a) Prove that

\displaystyle\lim_{x\to c^-}s(x)

and

\displaystyle\lim_{x\to c^+}s(x)

both exist for every real value of

c

. Your proof must include explicit calculations or limit laws for each one-sided limit. (2) Part 2: (b) Determine every real value of

c

for which

\displaystyle\lim_{x\to c}s(x)

exists and

s

is continuous at

x=c

. Provide complete justification, clearly referencing your results from part (a). (3) Part 3: (c) For all values of

c

that do not satisfy the condition found in part (b), classify the discontinuity at

x=c

and state the exact equation of any resulting vertical asymptote. (4) Part 4: (d) Take

c=-1

as a representative case where

s

possesses a vertical asymptote. Describe the precise end behavior of

s(x)

x\to -1^-

and as

x\to -1^+

. Include the sign (positive or negative) of the unbounded growth in each direction and relate this behavior to the algebraic structure of

s

. No numerical graph is required, but your analysis must be sufficiently detailed to allow another student to sketch the behavior near

x=-1