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SOME NUMERICAL EXAMPLES:
EXAMPLE 1:
Let's look at the sequence whose nth
term is given by n/(n+1). Recall, that we
let n=1 to get the first term of the
sequence, we let n=2 to get the second
term of the sequence and so on.
What will this sequence look like?
1/2,
2/3,
3/4,
4/5,
5/6,...
10/11,...
99/100,...
99999/100000,...
What's happening to the terms of
this sequence? Can you think of a number that these terms are getting
closer and closer to? Yep! The terms are getting closer to 1!
But, will they ever get to 1? Nope! So,
we can say that these terms are approaching 1.
Sounds like a limit! The limit is 1.
As n
gets bigger and bigger, n/(n+1) gets
closer and closer to 1...
EXAMPLE 2:
Now, let's look at the sequence
whose nth term is given by 1/n.
What will this sequence look like?
1/1,
1/2,
1/3,
1/4,
1/5,...
1/10,...
1/1000,...
1/1000000000,...
As n
gets bigger, what are these terms approaching? That's right! They are
approaching 0. How can we write this in
Calculus language?
What if we stick an x in for the n?
Maybe it will look familiar... Do you remember what the graph of
f(x)=1/x looks like? Keep reading to
see our second example shown in graphical terms!
Continued on the
next
page
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