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Limits Explained in 5 Minutes

Limits Explained in 5 Minutes

Consider this sequence of numbers..

1, 1/2, 1/3, 1/4, 1/5,..

It's pretty clear that this sequence goes on forever because there are an infinite number of integers. The general member of this sequence is 1/n where n is an integer, so we could write the sequence in a more compact form as {1/n} with the understanding that n goes from 1 to infinity.

Now as n gets bigger 1/n gets smaller. In fact, we can make 1/n as small as we please by simply making n big enough. In other words we could make 1/n come as close as we like to zero. The important phrase here is "as close as we like" because no matter how hard we try we can never make 1/n exactly zero!

This is the idea behind a limit. Mathematicians say the "limit" of 1/n as n goes to infinity is 0. So a limit is something we can approach "as close as we like" but we can never actually get there!

It's written like this..

Lim{1/n}=0 as n goes to infinity.

Another way to think about this is..

How do the terms of the sequence {1/n} behave as n goes towards infinity? The terms approach 0. So we say the limit is 0. Another terminology is to say that the sequence "converges" to 0 as n goes to infinity. So convergence becomes an interesting property of sequences, and we can ask questions like this..

Does a sequence converge, and if so to what limit?

This idea of a limit can be generalized. It does not just apply to sequences and integers.

For example, we could take a function f(x) and ask how it behaves as x approaches some value a. Again, the critical word here is "approaches" meaning x can get as close as we want to the value a but can never actually equal a. We're asking an interesting question..

How does f(x) behave in the immediate vicinity of the value x=a?

You may think the answer is obvious, that as x approaches a then f(x) must approach f(a). It turns out this is true for many functions but not all. So it's not obvious!

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Content written and posted by Ken Abbott