Mathematics

Problem

Consider the function

F(x)=\frac{x^{2}-4}{x-2},

which is undefined at

x=2

. (a) Express

F(x)

in an equivalent algebraic form that is defined for all

x\ne 2

and simplifies evaluation of the limit as

x\to 2

. (b) Using the form obtained in part (a), determine

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

. Show each algebraic step and cite any properties of limits you employ. (c) Decide whether

F

can be extended to a continuous function at

x=2

. If so, define

F(2)

so that the extended function is continuous and justify your answer.

Intuitive explanation

The key is recognizing that even though $F(x)$ has a "hole" at $x=2$ , you can still find what value the function approaches there by simplifying algebraically first. Factor the numerator as a difference of squares, cancel the common factor with the denominator, and you'll reveal the underlying linear function hiding inside. This limit tells you the $y$ -coordinate of that hole.

Intuitive explanation

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