A computer–generated plot of a function is described below

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

Problem

A computer–generated plot of a function

f

is described below. The

x

–axis is scaled in units of

1

, whereas the

y

–axis is compressed so that one vertical unit occupies only one–half of a horizontal unit. • Near

x=2

, the curve approaches the point

(2,3)

from both sides but the plotting software leaves a visible gap indicating that the point is missing. • The curve passes smoothly through

(0,1)

and

(4,5)

following the cubic trend

y=\tfrac12(x-2)^3+3

everywhere except exactly at

x=2

. • At

x=5

, the curve shoots upward steeply; zooming reveals that the curve behaves like

y=\dfrac{4}{(x-5)^2}-1

for

x\ne5

. • No other irregularities are seen on the domain

(-1,7)

. Parts: (a) Estimate

\displaystyle \lim_{x\to2}f(x)

and state

f(2)

as implied by the description. Classify the discontinuity at

x=2

and justify your answer in terms of limits. (b) Determine

\displaystyle \lim_{x\to5^-}f(x)

and

\displaystyle \lim_{x\to5^+}f(x)

. Classify the discontinuity at

x=5

and justify your answer. (c) Propose an explicit piecewise formula for

f

(-1,7)

that reproduces the described behavior at

x=2

and

x=5

. Clearly indicate any domain restrictions in your definition and explain how your formula matches the graphical features.