I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 2.5 : Computing Limits
15. Use the Squeeze Theorem to determine the value of \(\mathop {\lim }\limits_{x \to 0} {x^4}\sin \left( {\frac{\pi }{x}} \right)\).
We first need to determine lower/upper functions. We’ll start off by acknowledging that provided \(x \ne 0\) (which we know it won’t be because we are looking at the limit as\(x \to 0\)) we will have,
\[ - 1 \le \sin \left( {\frac{\pi }{x}} \right) \le 1\]Now, simply multiply through this by \({x^4}\) to get,
\[ - {x^4} \le {x^4}\sin \left( {\frac{\pi }{x}} \right) \le {x^4}\]Before proceeding note that we can only do this because we know that \({x^4} > 0\) for \(x \ne 0\). Recall that if we multiply through an inequality by a negative number we would have had to switch the signs. So, for instance, had we multiplied through by \({x^3}\) we would have had issues because this is positive if \(x > 0\) and negative if \(x < 0\).
Now, let’s get back to the problem. We have a set of lower/upper functions and clearly,
\[\mathop {\lim }\limits_{x \to 0} {x^4} = \mathop {\lim }\limits_{x \to 0} \left( { - {x^4}} \right) = 0\]Therefore, by the Squeeze Theorem we must have,
\[\mathop {\lim }\limits_{x \to 0} {x^4}\sin \left( {\frac{\pi }{x}} \right) = 0\]