Limits Involving Infinity – Limits and Continuity

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 … Read moreLimits Involving Infinity – Limits and Continuity

Computing Limits II: The Squeeze Theorem – Application Proof

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 … Read moreComputing Limits II: The Squeeze Theorem – Application Proof

Computing Limits I – Limits and Continuity

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 … Read moreComputing Limits I – Limits and Continuity

Introduction to Limits – Limits and Continuity

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\[ … Read moreIntroduction to Limits – Limits and Continuity