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

x��a

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

x��a

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

x��a

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

x��0

and

lim+ H(x) = 1.

x��0

For any function f and any a, the general limit limx��a 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.

x��a

x��a

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