Definition 1 - limits

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Definition 4
Let f(x)f(x) be a function defined on an interval that contains x=ax=a,
except possibly at x=ax=a. Then we say that,limx→af(x)=∞limx→a⁡f(x)=∞
if for every number M>0M>0 there is some number δ>0δ>0such that
f(x)>Mwhenever0<|x−a|<δf(x)>Mwhenever0<|x−a|<δ
Definition 5
Let f(x)f(x) be a function defined on an interval that contains x=ax=a,
except possibly at x=ax=a. Then we say
that,limx→af(x)=−∞limx→a⁡f(x)=−∞
if for every number N<0N<0 there is some number δ>0δ>0such that
f(x)<Nwhenever0<|x−a|<δf(x)<Nwhenever0<|x−a|<δ
In these two definitions note that MM must be a positive number and
that NN must be a negative number. That’s an easy distinction to miss if
you aren’t paying close attention.
Also note that we could also write down definitions for one-sided limits that
are infinity if we wanted to. We’ll leave that to you to do if you’d like to.
Here is a quick sketch illustrating Definition 4.
What Definition 4 is telling us is that no matter how large we choose MM to
be we can always find an interval around x=ax=a, given
by 0<|x−a|<δ0<|x−a|<δ for some number δδ, so that as long as we stay
within that interval the graph of the function will be above the
line y=My=Mas shown in the graph above. Also note that we don’t need
the function to actually exist at x=ax=a in order for the definition to hold.
This is also illustrated in the graph above.
Note as well that the larger MM is the smaller we’re probably going to need
to make δδ.
To see an illustration of Definition 5 reflect the above graph about the xxaxis and you’ll see a sketch of Definition 5.
Let’s work a quick example of one of these to see how these differ from the
previous examples.
Example 5 Use the definition of the limit to prove the following
limit.limx→01x2=∞limx→0⁡1x2=∞
Hide Solution
These work in pretty much the same manner as the previous set of
examples do. The main difference is that we’re working with an MM now
instead of an εε. So, let’s get going.
Let M>0M>0 be any number and we’ll need to choose a δ>0δ>0 so that,
1x2>Mwhenever0<|x−0|=|x|<δ1x2>Mwhenever0<|x−0|=|x|<δ
As with the all the previous problems we’ll start with the left inequality and
try to get something in the end that looks like the right inequality. To do this
we’ll basically solve the left inequality for xx and we’ll need to recall
that √ x2 =|x|x2=|x|. So, here’s that work.
1x2>M⇒x2<1M⇒|x|<1√ M 1x2>M⇒x2<1M⇒|x|<1M
So, it looks like we can choose δ=1√ M δ=1M. All we need to do now is verify
this guess.
Let M>0M>0 be any number, choose δ=1√ M δ=1M and assume
that 0<|x|<1√ M 0<|x|<1M.
In the previous examples we tried to show that our assumptions satisfied
the left inequality by working with it directly. However, in this, the function
and our assumption on xx that we’ve got actually will make this easier to
start with the assumption on xx and show that we can get the left inequality
out of that. Note that this is being done this way mostly because of the
function that we’re working with and not because of the type of limit that
we’ve got.
Doing this work gives,
|x|<1√ M |x|2<1Msquare both sidesx2<1Macknowledge that |x|2=x21x2>Msolve
for x2|x|<1M|x|2<1Msquare both sidesx2<1Macknowledge
that |x|2=x21x2>Msolve for x2
So, we’ve managed to show that,
1x2>Mwhenever0<|x−0|<1√ M 1x2>Mwhenever0<|x−0|<1M
and so by the definition of the limit we have,
limx→01x2=∞
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