Limits and rates of change

Limits investigate what happens as we approach a certain value, not exactly at that value. When \(x\) approaches \(c\), it gets progressively closer to \(c\) without actually reaching \(c\). We denote this by \(x\to c\). If \(f\) is a function of \(x\), its value \(f(x)\) may approach something in particular as \(x\to c\). We then say that the function has a limit as \(x\to c\). We denote this by \(\lim\limits_{x\to c}f(x)=L\).

If \(x\to c\) while being less than \(c\), we say that it approaches \(c\) ‘from the left’ or ‘from below’ and we denote this by \(x\to c^-\). If it approaches \(c\) while being greater than \(c\), we say that it approaches ‘from the right’ or ‘from above’ and we denote this by \(x\to c^+\). The limit as \(x\to c\) from one side only is called a one-sided limit.

The limit of a function exists if and only if both one-sided limits exist, and they are equal. \[\lim\limits_{x\to c}f(x)=L\iff \lim\limits_{x\to c^-}f(x)=\lim\limits_{x\to c^+}f(x)=L\]

The notation \(x\to\infty\) means that the value of \(x\) increases without bound. The notation \(x\to-\infty\) means it decreases without bound. Hence, the notation \(\lim\limits_{x\to c}f(x)=\infty\) means that \(f(x)\) increases without bound as \(x\to c\); and the notation \(\lim\limits_{x\to c}f(x)=-\infty\) means that \(f(x)\) decreases without bound as \(x\to c\).

If the value of \(x\) progressively approaches zero; i.e., \(x\to 0\), we say that \(x\) is an infinitesimal value (i.e., infinitely small value).