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Now we want to talk about calculating limits at infinity, so letting X approach infinity
or negative infinity, algebraically. So, if we're given the limit as X goes to infinity
of 1 over X, how would we figure that out? If X gets bigger and bigger and bigger, what
happens to 1 over X? If we do a little comparison here, when X is 10, 1 over X is one tenth.
If X is 100, it's one hundredth. If X is 1000, we get 1 thousandth. If X is a million, we
get one millionth. So, you can see that the bigger X gets, the smaller 1 over X gets.
1 over X appears to be approaching zero. So, that wasn't so hard to figure out. If we do
the same thing with X approaching negative infinity of 1 over X, so we plug in negative
10, we get negative one tenth, negative one hundred gives negative one hundredth, negative
one thousand gives us negative one thousandth, negative one million gives us negative one
millionth. So again, we're getting closer and closer to zero. It gets more complicated
if the function is more complicated. If we're asked to find something like the limit as
X goes to infinity of X over 5 X squared minus 11 times sine of X, now we have no idea what
to do. Off our head, probably it's infinity over infinity times who knows what. Before
we can solve problems like this, we need to investigate the properties of infinity more
closely. How do we work with infinity? It's not a number -- what is it? What are the rules
for doing some kind of reasoning with infinity?