One-sided limits provide a convenient means of describing how a function \(f\) behaves as its inputs approach a value \(a\) from one side or the other. We go straight to their formal definitions, mainly to introduce the notation and terminology.
Let \(f\) be a function, and let \(a\in \R\text{.}\)
Left-hand limit.
Assume \(f\) is defined on an open interval of the form \((c,a)\text{.}\) We say the left-hand limit of \(f\) as \(x\) approaches \(a\) from the left exists if there is a value \(L\) satisfying the following property: for all \(\epsilon > 0\text{,}\) there exists a \(\delta> 0\) such that if \(a-\delta< x< a\text{,}\) then \(\abs{f(x)-L}< \epsilon\text{.}\)
Assume \(f\) is defined on an open interval of the form \((a,d)\text{.}\) We say the right-hand limit of \(f\) as \(x\) approaches \(a\) from the right exists if there is a value \(L\) satisfying the following property: for all \(\epsilon > 0\text{,}\) there exists a \(\delta> 0\) such that if \(a< x< a+\delta\text{,}\) then \(\abs{f(x)-L}< \epsilon\text{.}\)
Informally we can describe left- and right-handed limits in much the same way as limits. We have \(\lim\limits_{x\to a^-}f(x)=L\) if we can make \(f(x)\) arbitrarily close to \(L\) for all \(x\) sufficiently close to, and lying to the left of \(a\text{.}\) An exactly analogous statement holds for the right-hand limit statement \(\lim_{x\to a^+}=L\text{.}\)
Neither \(\lim\limits_{x\to a^-}f(x)\) nor \(\lim\limits_{x\to a^+}f(x)\) exists, since the function values get arbitrarily large (and negative) as \(x\) approaches \(a\) from either side.
We have \(\lim\limits_{x\to d^-}f(x)=C\text{.}\) By contrast, \(\lim\limits_{x\to d^+}f(x)\) does not exist as the values of \(f\) oscillate wildly as \(x\) approaches \(d\) from the right.
Since the definition of a one-sided limit differs from that of the limit, only in so far as we restrict our attention to inputs \(x\) lying to one side or the other of the βlimit pointβ \(a\text{,}\) it should come as no surprise that all the limit rules (sum, quotient, sandwich, etc.) and formulas (polynomial evaluation, rational evaluation, etc.) discussed thus far apply with equal validity to one-sided limits. This means you may compute one-sided limits with these rules in exactly the same way as you do with normal limits.
Previously our definition of limit only applied to points \(a\in \R\) for which the function was defined for all \(x\ne a\) on an open interval \((c,d)\) containing \(a\text{.}\) The language of one-sided limits allows us to extend this definition to βendpointsβ of the domain, as the next definition makes clear. TheoremΒ 5.6 guarantees that this definition is a consistent extension of DefinitionΒ 4.1.
Let \(f\) be a function, and let \(a\in \R\text{.}\)
Right endpoint.
If \(f\) is defined for all \(x\) on an open interval \((c,a)\text{,}\) but not for all \(x\ne a\) in a full open interval containing \(a\text{,}\) then we say the limit of \(f\) as \(x\) approaches \(a\) exists if \(\lim\limits_{x\to a^-}f(x)\) exists. When this is the case, we define
If \(f\) is defined for all \(x\) on an open interval \((a,d)\text{,}\) but not for all \(x\ne a\) in a full open interval containing \(a\text{,}\) then we say the limit of \(f\) as \(x\) approaches \(a\) exists if \(\lim\limits_{x\to a^+}f(x)\) exists. When this is the case, we define
As was suggested by ExampleΒ 5.2 and made official by the next theorem, \(\lim_{x\to a}f(x)\) exists if and only if both one-sided limits exist and are equal.
Suppose \(f\) is a function defined everywhere on an open interval containing the point \(a\in \R\text{,}\) except possibly at \(a\) itself. The limit \(\lim\limits_{x\to a}f(x)\) exists if and only if the one-sided limits \(\lim\limits_{x\to a^-}f(x)\) and \(\lim\limits_{x\to a^+}f(x)\) exist and are equal. Using logical shorthand:
As a consequence, if one of the one-sided limits does not exist, or if both one-sided limits exist but are not equal to one another, then the limit \(\lim\limits_{x\to a}f(x)\) does not exist.
TheoremΒ 5.6 is particularly useful in that it gives us a means of potentially showing that a limit does not exist. The next classic example illustrates this nicely.
We compute the left and right limits of \(f\) at \(0\) separately. As you will see, the piecewise nature of the definition of \(\abs{x}\) will make these quite easy.
We thus see that the two one-sided limits exist, but are not equal to one another. We conclude, using TheoremΒ 5.6 that the limit of \(f\) at 0 does not exist.
The following classic limit formulas are proved using one-sided limits computed with the help of some lovely geometric arguments and the sandwich theorem.
Our second computation of the limit in ExampleΒ 5.9 seemed to use a βsubstitutionβ rule of the following sort: that is, we computed a limit of the form \(\lim_{x\rightarrow a}f(g(x))\) by observing that \(\lim_{x\to a}g(x)=b\) and then computing, using the substitution \(t=g(x)\text{,}\) that
Intuitively this seems straightforward, since as \(x\) approaches \(a\text{,}\)\(t=g(\theta)\) approaches \(b\text{.}\) But in fact the claim is simply not true in general! Luckily for us, the approach does work when the βsubstitution functionβ is of the form \(g(x)=cx+d\text{,}\) a so-called affine function. As such we will use such methods without further comment.