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NARRATOR: Exercise 7.

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Examine whether or not
the function f of x

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equals 4 minus x squared
if x is less than or equal to 0

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or 4 plus x if x is greater than 0
is continuous at x equals 0.

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We first consider the limit
as x approaches 0

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from both above and below.

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As x approaches 0 from below,

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we'll be dealing with the part of the
function with rule 4 minus x squared.

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And so the limit as x approaches 0
from below is in fact 4.

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Thinking about the limit
as x approaches 0 from above,

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we'll be dealing with the part of
the function with rule 4 plus x.

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And so the limit as x approaches 0
from above is also 4.

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And so putting
these two limits together,

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since both the limit
as x approaches 0 from below

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and the limit as x approaches 0 from
above are equal to the same value,

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we say that the limit of f of x
as x approaches 0 exists

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and it does in fact equal 4.

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We also note that the value
of the function at 0 is also 4,

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so the function is defined at 0

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and there is also a limit
as x approaches 0.

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And more importantly than that,
since those two things are the same,

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that is since the limit
of f of x as x approaches 0

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is equal to the value
of the function at 0,

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we can say that f of x is continuous
at x equals 0.

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Consideration of the graph

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of y equals f of x

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also allows us to informally

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determine continuity at x equals 0.

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Since the graph of y equals f of x

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can be drawn through x equals 0

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without lifting

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the pen from the paper,

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we can say that f of x is continuous

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at x equals 0.