2.2. Limits of Functions - University of Manitoba
2.2. Limits of Functions
We now start the real content of the course.
A limit is a tool for describing how (real-valued) functions behave
close to a point.
We write
lim f (x) = L
xa
and say
the limit of f (x), as x approaches a, equals L,
if the values of f (x) can be made as close as we like to L by taking
x to be su?ciently close to a (on either side of a) but not equal
to a.
A limit involves what is going on around a point, and does not
care what happens at it.
We will talk more soon about how to calculate the limits exactly.
Guessing is never the way to go in practice, but if you work through
the guessing examples in the book, you may have a better idea of
what is going on and how limits work.
One-sided Limits
Write
lim? f (x) = L
xa
and say that the left-hand limit of f (x) as x approaches
a is equal to L if we can make the values of f (x) arbitrarily close
to L by taking x to be su?ciently close to a and x less than a.
Write
lim+ f (x) = L
xa
and say that the right-hand limit of f (x) as x approaches
a is equal to L if we can make the values of f (x) arbitrarily close
to L by taking x to be su?ciently close to a and x greater than a.
So,
lim? H(x) = 0
x0
and
lim+ H(x) = 1.
x0
For any function f and any a, the general limit limxa f (x) exists
and equals L if and only if both the left-hand the right-hand limits
exist, and
lim? f (x) = lim+ f (x) = L.
xa
xa
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