For the given function, ƒ, determine if the function is discontinuous at a given point. State which condition of continuity fails or state the function is conti

Question

For the given function, ƒ, determine if the function is discontinuous at a given point. State which condition of continuity fails or state the function is continuous at x = a.

 

f(x) = (x^2 - 4)/(x - 2), a = 2

Answer 1

Ok, you can rewrite the function so that it is

 

(x + 2)(x - 2)

----------------

   (x - 2)

 

which when you simplify leaves you with 

 

f(x) = x + 2 

 

so you might THINK that the equation is continuous all the time because you get a value for every x, but you would be wrong.

 

Look back at the original function and notice that the denominator is 

x - 2.

 

You know from fractions that the denominator must NEVER be 0, it causes fires to start and rips a hole in a distant galaxy.  Because of that the function will have an hole where it approaches, but never reaches 2, and is therefore discontinuous.