Computing Limits I – Limits and Continuity

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 … 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

Computing Limits II: The Squeeze Theorem – Limits and Continuity

The Squeeze Theorem Let’s start with a statement of the squeeze theorem. Theorem 1.1  Suppose \(g(x) \le f(x) \le h(x)\) and this is true for all \(x\) in a neighbourhood of \(x = a\) (except perhaps at \(x = a\)). Further, suppose \(\displaystyle \lim_{x \to a} g(x) = \displaystyle \lim_{x \to a} h(x) = L\). Then … Read moreComputing Limits II: The Squeeze Theorem – Limits and Continuity