The graph of a function is described verbally as follows

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The graph of a function

f

is described verbally as follows. • For

x<2

, the graph is a smooth curve that approaches the point

(2,3)

from the left. • For

x>2

, the same smooth curve approaches the point

(2,3)

from the right. • The point

(2,3)

is not plotted on the graph; instead, a filled dot appears at

(2,5)

. Answer the following questions about

f

. (a) Determine

\displaystyle \lim_{x\to 2}f(x)

and state its numerical value. (b) Explain why

f

is not continuous at

x=2

even though the two one‐sided limits you found in part (a) exist and are equal. (c) Describe how the value of

f

x=2

must be defined (or redefined) so that the resulting function becomes continuous at

x=2

. Give a clear verbal explanation referencing the limit and the corresponding point that should appear on the graph.