## 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  Limits 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\[ […]