Consider the piecewise function P(x)= cases x^ 2 -1 x-1 , & x<1

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

Problem

Consider the piecewise function

P(x)=\begin{cases} \dfrac{x^{2}-1}{x-1}, & x<1,\\ \dfrac{m}{x-1}+5, & x\ge 1, \end{cases}

where

m

is a real constant. Analyze how the value of

m

influences the limit of

P

x=1

and the corresponding continuity or discontinuity of

P

. (a) Evaluate

\displaystyle\lim_{x\to 1^-}P(x)

. (b) For a general real value of

m

, determine

\displaystyle\lim_{x\to 1^+}P(x)

or explain why the limit does not exist. (c) Decide whether there exists any real value of

m

for which

\displaystyle\lim_{x\to 1}P(x)

exists and is finite. If such a value exists, state it and fully justify your conclusion; if no such value exists, prove why not. (d) For all values of

m

that do not satisfy the condition found in part (c), classify the discontinuity of

P

x=1

and explicitly state the equation(s) of any vertical asymptote(s).

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