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Calculus/Limit problem

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Expert: - 10/8/2009


Question
QUESTION: f(x)=  (x-3)^2 if x > 1

   =   25 if x <1

if lim x -> a f(x) exists what is a?

lim f(x) -> a+ = lim f(x) ->a-

ANSWER: The value for a could be anything but 1 for the limit to exist.

For x=1, the +limit would be (1-3)² = 4.

For x=1, the -limit would be 25.


---------- FOLLOW-UP ----------

QUESTION: sorry I didnt write the question correctly

f(x)=  (g(x)-3)^2 if x > 1

  =   25 if x <1
if lim x -> a f(x) exists what is the value of g(x)?

lim f(x) -> a+ = lim f(x) ->a-
and the limit always exist even as x >1

Answer
The problem to be looked at is what is g(1).

If f(1) = 25, the function is continuous.

Now f(1) = (g(1)-3)².  For (g(1)-3)² to be 25, take the squareroot of both sides.
It can be seen we need g(1) - 3 = ±5.

This means that g(1) = 3±5.  The two values for 3±5 are -2 and 8.

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