Consider the function

As x gets closer to 2 from above we notice that f(x) gets as close as we like to 1. We say that f has limit 1 as it tends to 2 from above (some people say from the right).

We write or, as .

Note that the value of f at x=2 is not relevant, for the concept of limits we do not even need x=2 to be in the domain for f.

In the same spirit,

.

Consider the function

As we get closer to 0 from above f(x) gets as large as you like, we say f(x) tends towards infinity.

We write, , or

as .

It does not mean that f(x) gets close to infinity. The value of f at x=0 is again irrelevant.

In the same spirit we have, .

If for a function f(x) and a point x=a we have

then we say that f has a limit h as x tends to a, and we write or as .

Likewise if we write simply .

For example, .

Consider the function .

Then

As x gets large and positive we see that f(x) gets as close to 2 as we like.

We write or as .

Likewise or as .

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