NARRATOR: Exercise 7.
Examine whether or not the function f of x
equals 4 minus x squared if x is less than or equal to 0
or 4 plus x if x is greater than 0 is continuous at x equals 0.
We first consider the limit as x approaches 0
from both above and below.
As x approaches 0 from below,
we'll be dealing with the part of the function with rule 4 minus x squared.
And so the limit as x approaches 0 from below is in fact 4.
Thinking about the limit as x approaches 0 from above,
we'll be dealing with the part of the function with rule 4 plus x.
And so the limit as x approaches 0 from above is also 4.
And so putting these two limits together,
since both the limit as x approaches 0 from below
and the limit as x approaches 0 from above are equal to the same value,
we say that the limit of f of x as x approaches 0 exists
and it does in fact equal 4.
We also note that the value of the function at 0 is also 4,
so the function is defined at 0
and there is also a limit as x approaches 0.
And more importantly than that, since those two things are the same,
that is since the limit of f of x as x approaches 0
is equal to the value of the function at 0,
we can say that f of x is continuous at x equals 0.
Consideration of the graph
of y equals f of x
also allows us to informally
determine continuity at x equals 0.
Since the graph of y equals f of x
can be drawn through x equals 0
without lifting
the pen from the paper,
we can say that f of x is continuous
at x equals 0.