Archive for the ‘Lectures’ Category

Limits Involving Infinity – Limits and Continuity

Posted: 9th February 2013 by seanmathmodelguy in Lectures

Limits Involving Infinity Let’s start with what we mean when we say \(\displaystyle \lim_{x \to\infty} f(x) = L\) or \(\displaystyle \lim_{x \to-\infty} f(x) = L\). We say \(f(x)\) has limit \(L\) as \(x\) approaches infinity (\(\infty\)) and write \(\displaystyle \lim_{x \to\infty} f(x) = L\) if as \(x\) moves increasingly far from the origin in the […]

Computing Limits II: The Squeeze Theorem – Application Proof

Posted: 9th February 2013 by seanmathmodelguy in Lectures

One very important application of the squeeze theorem is the proof that \(\displaystyle \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\). We present this proof next. This proof is for the limit as \(\theta\to 0^+\). The case for \(\theta\to 0^-\) can be proved in exactly the same manner and we leave it as an exercise to […]

Computing Limits I – Limits and Continuity

Posted: 29th November 2012 by seanmathmodelguy in Lectures

Computing ┬áLimits I In order to compute a limit algebraically, one needs to know what is and more importantly what is not allowed when manipulating limits. For example consider computing\[ \lim_{x\to 1}\frac{x^2-1}{x-1}. \]One cannot just substitute \(x=1\) into this expression because doing so results in the indeterminate form \(0/0\). By factoring the top of the […]

Introduction to Limits – Limits and Continuity

Posted: 13th October 2012 by seanmathmodelguy in Lectures

Introduction to Limits The limit of a function is concerned with the behaviour of a function near a given point. What happens at the point is of no concern. To be more precise, if “the limit of the function \(f(x)\) as \(x\) approaches the point \(a\)” is the value \(L\) then this is denoted as\[ […]