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