Video Transcript
Use the squeeze theorem to evaluate
the limit of two 𝜃 squared cos of one over 𝜃 as 𝜃 approaches zero.
Let’s recall the squeeze
theorem. The squeeze theorem says that if
there exist functions 𝑓 of 𝑥, 𝑔 of 𝑥, and ℎ of 𝑥 such that 𝑔 of 𝑥 is greater
than or equal to 𝑓 of 𝑥 but less than or equal to ℎ of 𝑥 when 𝑥 is near 𝑎,
except possibly at 𝑎. And the limit as 𝑥 approaches 𝑎
of 𝑓 of 𝑥 equals the limit as 𝑥 approaches 𝑎 of ℎ of 𝑥 equals 𝐿. Then the limit as 𝑥 approaches 𝑎
of 𝑔 of 𝑥 equals 𝐿.
In order to use the squeeze
theorem, we need to find a function 𝑔 of 𝑥 that we can bound from above and below
by functions ℎ of 𝑥 and 𝑓 of 𝑥, respectively. The limit we are asked to evaluate
in the question contains the function cos of one over 𝜃. Note that the value cos of 𝑥 is
always less than or equal to one and greater than or equal to negative one for any
of real number 𝑥. And so the function cos of one over
𝜃 is bounded from above by the constant function one and bounded from below by the
constant function negative one.
Now let’s multiply these
inequalities by two 𝜃 squared. Doing so, we obtain two 𝜃 squared
cos of one over 𝜃, which is the function whose limit we’re asked to evaluate as 𝜃
approaches zero, is less than or equal to two 𝜃 squared and greater than or equal
to negative two 𝜃 squared. Note that these bounds hold for any
real number 𝜃 near zero apart from 𝜃 equals zero, as one over 𝜃 is not defined
when 𝜃 is equal to zero.
As a result, letting the function
𝑔 equal to 𝜃 squared cos of one over 𝜃, the function 𝑓 equal negative two 𝜃
squared, the function ℎ equal two 𝜃 squared, and 𝑎 equal zero in the squeeze
theorem. We find that if the limit of
negative two 𝜃 squared as 𝜃 approaches zero equals the limit of two 𝜃 squared as
𝜃 approaches zero equals 𝐿. Then the limit of two 𝜃 squared
cos of one over 𝜃 as 𝜃 approaches zero also equals 𝐿.
Let’s see if we can evaluate the
limit as 𝜃 approaches zero of negative two 𝜃 squared and the limit as 𝜃
approaches zero of two 𝜃 squared. Firstly, note that we can factor
the multiplicative constant out of a limit. Therefore, the limit of negative
two 𝜃 squared as 𝜃 approaches zero is equal to negative two multiplied by the
limit of 𝜃 squared as 𝜃 approaches zero. For the same reason, the limit of
two 𝜃 squared as 𝜃 approaches zero is equal to two multiplied by the limit of 𝜃
squared as 𝜃 approaches zero.
Now let’s evaluate the limit as 𝜃
approaches zero of 𝜃 squared. In order to do this, we will use
the fact that the limit of 𝑥 to the power of 𝑛 as 𝑥 approaches 𝑎 is equal to 𝑎
to the power of 𝑛 for all positive integers 𝑛. Substituting 𝑎 equals zero and 𝑛
equals two into this result, we find that the limit of 𝜃 squared as 𝜃 approaches
zero is equal to zero squared, which is just zero.
We therefore find that the limit of
negative two 𝜃 squared as 𝜃 approaches zero is equal to negative two multiplied by
zero, which equals zero. And the limit of two 𝜃 squared as
𝜃 approaches zero is equal to two multiplied by zero, which also equals zero. So the limit of negative two 𝜃
squared as 𝜃 approaches zero is equal to the limit of two 𝜃 squared as 𝜃
approaches zero. And they are both equal to
zero.
Therefore, by the squeeze theorem,
the limit of two 𝜃 squared cos of one over 𝜃 as 𝜃 approaches zero is also equal
to zero, which is our final answer.
