Archive for February, 2013

Intermediate Value Theorem – Limits and Continuity

Posted: 12th February 2013 by seanmathmodelguy in Lectures

Intermediate Value Theorem To begin with, let’s start with the basic statement of the theorem. Theorem If \(f(x)\) is continuous on a closed interval \([a,b]\) and \(N\) is any number \(f(a) < N < f(b)\) then there exists a value \(c \in (a,b)\) such \(f(c) = N\). The illustration corresponding to the theorem is to […]

Continuity – Limits and Continuity

Posted: 12th February 2013 by seanmathmodelguy in Lectures

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 […]

Strategy to Calculate Limits – Limits and Continuity

Posted: 9th February 2013 by seanmathmodelguy in Lectures

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 […]

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 […]