Continuity – Limits and Continuity

Continuity A function \(f(x)\) is said to be continuous at a point \(a\) in its domain if the following three properties hold. \(\displaystyle \lim_{x \to a} f(x)\) exists. This takes three steps to show in itself. \(f(a)\) has to exist, \(\displaystyle \lim_{x \to a} f(x) = f(a)\). Continuity connects the behaviour of a function in … Read moreContinuity – Limits and Continuity

Strategy to Calculate Limits – Limits and Continuity

Strategy to Calculate Limits   To compute \(\displaystyle \lim_{x \to a} f(x)\): A. Try to plug the value of \(a\) directly into the function. If we get a number or the limit ‘blows up’ then we are done! You should be so lucky. Typically the value is undefined, having the form \(\displaystyle \frac{0}{0}\) or \(\displaystyle … Read moreStrategy to Calculate Limits – Limits and Continuity

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