### Video Transcript

Calculate the limit of π₯ squared
times cos of two over π₯ as π₯ approaches zero using the squeeze theorem.

Letβs remind ourselves of the
squeeze theorem. It says that if π of π₯ is less
than or equal to π of π₯ which is less than or equal to β of π₯ when π₯ is near π
and the limit of π of π₯ equals the limit of β of π₯ as π₯ approaches π. Weβll call this limit πΏ. Then the limit of π of π₯ as π₯
approaches π is also πΏ. Now how do we use this theorem to
calculate this limit here? Well, if we let π of π₯ equal π₯
cubed times cos of two over π₯ and we let π equal zero, then the limit we have to
find is the limit of π of π₯ as π₯ approaches π.

Now if we can find functions π and
β which underestimate and overestimate the function π, respectively, and which have
the same limit as π₯ approaches π which is zero. Then the limit weβre looking for
will also have this value. We write this idea down to remind
ourselves of it. Now all we have to do is find an
underestimate π of π₯ and an overestimate β of π₯ for our function which have the
same limit as π₯ approaches zero. How are we going to do this?

Well, the difficult part of our
function is the factor cos two over π₯. This is the bit of the function
that means that we canβt just do direct substitution. In fact, the limit of cos of two
over π₯ as π₯ approaches zero is undefined you can see this by graphing the function
using a computer or graphing calculator. However, there are no asymptotes
here. The range of cosine is from
negative one to one. The cosine of any number will have
to lie between negative one and one. And that remains true even that
number is two over π₯. Another way of saying this is that
the absolute value of cosine of two over π₯ is always less than or equal to one. As a result, the absolute value of
our function π of π₯ is less than or equal to the absolute value of π₯ cubed. Another way of saying this is that
π₯ cubed cos two over π₯ is between negative the absolute value of π₯ cubed and the
absolute value of π₯ cubed.

Have we found our functions π of
π₯ and β of π₯ then? Well, maybe. But we need to check that their
limits as π₯ approaches zero are the same. Letβs clear some room to do
this. We need to find the limit of π of
π₯ β thatβs negative the absolute value of π₯ cubed as π₯ approaches zero β and the
limit of β of π₯. Thatβs the absolute value of π₯
cubed as π₯ approaches zero. Both of these functions are
continuous. And so, these limits can be
evaluated using direct substitution. Negative the absolute value of zero
cubed is just zero as is the absolute value of zero cubed. So yes, the limits of π of π₯ and
β of π₯ as π₯ approaches zero are the same.

The limit value πΏ is zero. And so, by the squeeze theorem, the
limit of π₯ cubed times cos of two over π₯ as π₯ approaches zero is also zero. This is our answer. Looking at this diagram, you can
see how the graph of the function π₯ cubed times cos of two over π₯ is squeezed
between the graphs of the absolute value of π₯ cubed and itβs opposite. And so, its limit as π₯ approaches
zero must be zero, although the function itself is not defined at π₯ equals
zero.