A function is represented graphically on the interval

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

Problem

A function

f

is represented graphically on the interval

0\le x\le 6

. The graph may be described verbally as follows. • For

0\le x<2

, the graph is a line segment with positive slope that ends at the open point

(2,3)

. • For

2\le x<2.5

, the graph is a horizontal segment at height

y=-1

beginning with a filled point at

(2,-1)

and ending with an open point at

(2.5,-1)

. • For

2.5\le x<4

, the graph follows a curve that approaches

y=\infty

x\to 3^{+}

and then approaches

y=0

x\to 4^{-}

. The curve includes no breaks on

(2.5,4)

. • At

x=4

, there is no point on the graph. • For

4<x\le 6

, the graph is defined and continuous, re-entering from below at

(4,2)

and increasing smoothly to

(6,5)

. Answer the following questions about the behavior of

f

at the specified

x

-values. (a) Determine the values of

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

and

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

. Does

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

exist? Explain your reasoning and classify the specific reason for any failure of the limit to exist at

x=2

. Use precise limit terminology in your explanation. (b) Describe the behavior of

f

x

approaches

3

from the right. Based on this behavior, determine whether

\displaystyle\lim_{x\to3^{+}}f(x)

exists as a finite real number. Justify your answer and classify the reason for non-existence if applicable. (c) Using both one-sided limits, evaluate

\displaystyle\lim_{x\to4}f(x)

. Provide a detailed justification, including an explanation of the graphical evidence that supports your conclusion and the classification of any limit failure.

Related theory

Browse the problem library