Limits part 2: Functions: All kinds of limits
One-sided limits
Consider the function \[ f(x) = \frac{x^2-1}{|x-1|}\text. \] This function is defined for all \(x\neq 1\), so it is interesting to examine what happens around \(x=1\). In the table below we fill in \(x\) values near \(1\):
| \(x\) | 0.9 | 0.99 | 0999 | 0.9999 | 1 | 1.0001 | 1001 | 1:01 | 1.1 |
| \(f(x)\) | -1.9 | -1.99 | -1 999 | -1.9999 | 2.0001 | 2001 | 2:01 | 2.1 |
We see that the function values to the left of the table approach \(-2\), while on the right side of \(1\) it seems that the limit is \(2\).
The left limit of \(f\) at a point \(a\) is \(L\) if for every number \(\epsilon > 0\) there exists a number \(\delta>0\) such that if \(a-\delta < x < a\) then \(\left| f(x) - L \right|<\epsilon\).
In other words, for \(x\in \left( a - \delta, a\right)\) the function value \(f(x)\) is close to \(L\). We write this in symbols as \[ \lim_{x \uparrow a} f(x) = L\text.\] Another conventional notation for the left limit is \(\displaystyle \lim_{x\to a^-} f(x)\text.\) Similarly, we define the right limit of a function at a point.
The right limit of \(f\) at a point \(a\) is \(L\) if for every number \(\epsilon > 0\) there exists a number \(\delta>0\) such that if \(a < x < a + \delta\) then \(\left| f(x) - L \right|<\epsilon\).
We denote this as \[ \lim_{x \downarrow a} f(x) = L\text. \] Another conventional notation for right limit is \(\displaystyle\lim_{x\to a^+} f(x)\text.\)
In our notation, the example can be written as \[ \lim_{x\uparrow 1} \frac{x^2-1}{|x-1|} = -2 \quad\text{ and }\quad \lim_{x\downarrow 1} \frac{x^2-1}{|x-1|} = 2\]
