Video: Using the Squeeze Theorem

Calculate lim_(π‘₯ β†’ 0) π‘₯Β³ cos (2/π‘₯) using the squeeze theorem.

03:35

Video Transcript

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

Let’s remind ourselves of the squeeze theorem. It says that if 𝑓 of π‘₯ is less than or equal to 𝑔 of π‘₯ which is less than or equal to β„Ž of π‘₯ when π‘₯ is near π‘Ž and the limit of 𝑓 of π‘₯ equals the limit of β„Ž of π‘₯ as π‘₯ approaches π‘Ž. We’ll call this limit 𝐿. Then the limit of 𝑔 of π‘₯ as π‘₯ approaches π‘Ž is also 𝐿. Now how do we use this theorem to calculate this limit here? Well, if we let 𝑔 of π‘₯ equal π‘₯ cubed times cos of two over π‘₯ and we let π‘Ž equal zero, then the limit we have to find is the limit of 𝑔 of π‘₯ as π‘₯ approaches π‘Ž.

Now if we can find functions 𝑓 and β„Ž which underestimate and overestimate the function 𝑔, respectively, and which have the same limit as π‘₯ approaches π‘Ž which is zero. Then the limit we’re 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 𝑓 of π‘₯ and an overestimate β„Ž 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 can’t 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 𝑔 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 𝑓 of π‘₯ and β„Ž of π‘₯ then? Well, maybe. But we need to check that their limits as π‘₯ approaches zero are the same. Let’s clear some room to do this. We need to find the limit of 𝑓 of π‘₯ β€” that’s negative the absolute value of π‘₯ cubed as π‘₯ approaches zero β€” and the limit of β„Ž of π‘₯. That’s 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 𝑓 of π‘₯ and β„Ž of π‘₯ as π‘₯ approaches zero are the same.

The limit value 𝐿 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 it’s opposite. And so, its limit as π‘₯ approaches zero must be zero, although the function itself is not defined at π‘₯ equals zero.

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