Tip:
Highlight text to annotate it
X
We are looking at limits that do not exist and in this segment we are going to look at
limits that do not exist because their answer is a type of infinity. So for example, if
we look at the limit as x goes to 0 of one divided by x squared, we can see that if we
plug in 1 to that, we get one over one squared, which is just 1. If we plug in one half, we
get one divided by one half squared which is going to give us four. If we plug in one-tenth,
we get one over one-tenth squared which is going to be 100. If we plug in .001, we're
going to get 1 over .001 squared which is equal to.. that's going to give us a million.
And if we plug in -.001, the negative sign, when we square the bottom, is going to become
positive so it's also going to give us a million. And so what it looks like here from plugging
in nearby values is, the smaller number we plug in, the closer we get to 0, the bigger
one over x squared becomes. And so what we want to say is, that this limit goes to infinity
because it get bigger and bigger and bigger. Now infinity is not a number so the limit
doesn't exist but saying infinity is more specific than saying it doesn't exist. So,
if we are making a Venn diagram here of all the kinds of limits that do not exist, limits
that equal infinity and limits that equal negative infinity are special kinds of 'do
not exist', meaning that you want to actually give, if appropriate, the answer of infinity
or the answer of negative infinity but for any kinds of theorems or any kinds of rules
that we have but limits not existing, these count as kinds of non-existing. And if we look at the graph
of our function we can see that yes, it looks like it's going up to infinity.