Question

For the following exercises, determine why the function f is discontinuous at a given point a on the graph. State which condition fails. f(x) = 25 – x2/x2 – 10x + 25 

Answer 1

A function f (x) is continuous at x = a only if it satisfies all the three following conditions:

• Condition 1: f (a) exists.

• Condition 2: limx→af(x) exists

• Condition 3: limx→af(x) = f(a)

 

Consider the following function:

f(x) = 25 – x2/x2 – 10x + 25

 

Determine whether it is continuous for a = 5

f(a) = f(5)

       = [25 – x2/x2 – 10x + 25]x=5

       = [25 – x2/(x – 5)2]x=5

       = 0/0

 

Since 0/0 does not exist, so the function f(x) = 25 – x2/x2 – 10x + 25 fails to satisfy the condition 1 at a = 5.

So the function f(x) = 25 – x2/x2 – 10x + 25 is discontinuous at a = 5.

So the function f(x) = 25 – x2/x2 – 10x + 25 is discontinuous at a = 5.