A student sketches an unknown function on with the following visual features: • A…

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

Problem

A student sketches an unknown function

p

(-\infty,\infty)

with the following visual features: • A removable discontinuity (hole) at

x=-2

with a missing height of

3

. • A jump discontinuity at

x=1

where the left-hand limit equals

2

and the right-hand limit equals

5

. • A vertical asymptote at

x=4

with

p(x)\to -\infty

x\to 4^{-}

and

p(x)\to +\infty

x\to 4^{+}

. • Everywhere else, the graph is smooth and passes through the point

(0,0)

with positive slope there. Design an explicit algebraic formula for a function

p

that matches all of the graphical features listed above. (a) Construct a formula for

p(x)

that has the stated behaviors at

x=-2,1,4

and satisfies

p(0)=0

. Clearly indicate any domain restrictions and removable factors. (b) Justify analytically that your

p(x)

possesses each of the three discontinuities in the manner described. Your justification must include limit calculations that confirm the left-hand and right-hand behaviors as well as evaluations showing the removable nature of the discontinuity at

x=-2

. (c) Identify all open intervals on which your

p(x)

is continuous, and explain how your formula and limits demonstrate continuity on those intervals.

Related theory

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