Left and right limits

Sometimes we are interested in the behavior of a function only at one side of a point, like if the point is an endpoint of an interval we consider. We therefore introduce the concepts of left and right limits.

Just as for two-sided limits we both have an informal and a formal definition of left and right limits.

Definition An informal definition of left and right limits
We say that \(L\) is the left limit of the function \(f\) at a point \(a\) if we can get \(f(x)\) as close as we want to \(L\) by taking \(x\) to the left of \(a\) and close to \(a\), but not equal to \(a\).

We write \[\lim_{x\to a^-}f(x)=L\]

Similarly, we say that \(L\) is the right limit of the function \(f\) at a point \(a\) if we can get \(f(x)\) as close as we want to \(L\) by taking \(x\) to the right of \(a\) and close to \(a\), but not equal to \(a\).

We write \[\lim_{x\to a^+}f(x)=L\]

The informal definition of left and right limits given above is enough to get an intuitive understanding of limits, and in some cases compute them, but in general it is far to imprecise to work with. The following definition formalizes the above description of left and right limits.

Definition: A formal definition of left and right limits
Let \(f\) be a function defined on an interval \((b,a\)\). We say that \[L=\lim_{x\to a^-} \ f(x)\] provided that, for every \(\epsilon\) >0, there exists a \(\delta > 0\) such that for all \(x\), \[0 < (a-x) < \delta \implies | f(x) - L | < \epsilon\]

Similarly, if \(f\) is a function defined on an interval \((a,b\)\). We say that \[L=\lim_{x\to a^+} \ f(x)\] provided that, for every \(\epsilon\) >0, there exists a \(\delta > 0\) such that for all \(x\), \[0 < (x-a) < \delta \implies | f(x) - L | < \epsilon\]

Even when the left and the right limit of a function \(f\) both exist at a point \(a\), then the do not have to be equal: in fact, they are equal if and only if \(\lim_{x\to a}f(x)\) exists and is equal to \(\lim_{x\to a^-}f(x)\) and \(\lim_{x\to a^+}f(x)\).

Theorem: Relationship between one-sided and two-sided limits
A function \(f\) has limit \(L\) as \(x\) approaches \(a\) if and only if \(f(x)\) has both a left and a right limit as \(x\) approaches \(a\) and these one-sided limits both equal \(L\).

That is: \(\lim_{x\to a}f(x)=L\iff \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\).

Just as for two-sided limits, there are rules that are useful for computing one-sided limits.

Theorem: Rules for computing left and right limits
Suppose \(\lim_{x\to a^-}f(x)=L\), \(\lim_{x\to a^-}g(x)=M\) and \(k\) is a constant. Then the following rules for computing limits hold

1. \(\lim_{x\to a^-}[f(x)+g(x)]=L+M\)
2. \(\lim_{x\to a^-}[f(x)-g(x)]=L-M\)
3. \(\lim_{x\to a^-}f(x)g(x)=LM\)
4. \(\lim_{x\to a^-}kf(x)=kL\)
5. \(\lim_{x\to a^-}\frac{f(x)}{g(x)}=\frac{L}{M}, \quad \text{if} \,\, M\neq 0.\)
6. Furthermore, if \(m\) is an integer and \(n\) is a positive integer, then \(\lim_{x\to a^-}[f(x)]^{m/n}=L^{m/n}\), if \(L>0\) when \(n\) is even, and \(L\neq 0\) if \(m<0\).
7. If there is a \(c>0\) such that \(f(x)\le g(x)\) for all \(x\in (a-c,a)\cup (a,a+c)\), then \(L\le M\).

(Notice that we in 7. cannot conclude that \(L<M\) if \(f(x)< g(x)\) for all \(x\in (a-c,a)\cup (a,a+c)\).)

These rules also hold for left limits and right limits.

Another rule that is useful for computing limits is the Squeeze Theorem.

Theorem 4: The Squeeze Theorem
If there is a \(c>0\) such that \(f(x)\le g(x)\le h(x)\) for all \(x\in (a-c,a)\cup (a,a+c)\), and \(\lim_{x\to a}f(x)=\lim_{x\to a}h(x)=L\), then \(\lim_{x\to a}g(x)=L\).

This rule also holds for left limits and right limits.