Algebraic Laws For One-Sided Limits

Why One-Sided Limit Laws Need Care

One-sided limits let us study a function as its input approaches a point from only one direction. Once the direction is fixed, familiar algebraic operations often behave just as they do for two-sided limits. But the laws have conditions: both component limits must be taken from the same side, a quotient requires a nonzero denominator limit, and a composition must approach the outer input from the appropriate side.

These conditions matter even when an expression looks simple. A limit law is a rule for combining limits that are known to exist; it is not permission to manipulate any expression by substituting values without checking the assumptions.

One-Sided Limits and the Algebraic Laws

The notation \(x\to a^+\) means that \(x\) approaches \(a\) through values greater than \(a\). The notation \(x\to a^-\) means that \(x\) approaches \(a\) through values less than \(a\). A statement such as \(\lim_{x\to a^+}f(x)=L\) means that \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently close to \(a\) from the right. The left-hand version has the same meaning, with \(x<a\).

Here \(a^\pm\) is shorthand for either one chosen direction: all limits in a calculation must use the same direction. The laws do not require \(f\) or \(g\) to be defined at \(a\); they require the relevant expressions to be defined for inputs sufficiently near \(a\) on that side.

Why the Laws Work

The justification follows from the definition of a limit. We give the main estimates for either chosen side; the argument uses only inputs on that side.

For the sum, suppose \(f(x)\to L\) and \(g(x)\to M\). Given \(\varepsilon>0\), take \(x\) close enough to \(a\) that \(|f(x)-L|<\varepsilon/2\) and \(|g(x)-M|<\varepsilon/2\). Then

This proves the sum law. For the product, first choose \(x\) close enough that \(|g(x)-M|<1\), which gives \(|g(x)|<|M|+1\). Also make \(x\) close enough that \(|f(x)-L|\) and \(|g(x)-M|\) are both less than \(\varepsilon/[2(|M|+|L|+1)]\). The identity \(fg-LM=g(f-L)+L(g-M)\) then gives

Thus the product tends to \(LM\). For the quotient, \(g(x)\to M\ne0\) implies that, sufficiently near \(a\), \(|g(x)-M|<|M|/2\). Consequently \(|g(x)|>|M|/2\), so the quotient is defined there. Moreover,

The right-hand side tends to zero, so \(1/g(x)\to1/M\). Applying the product law to \(f(x)\cdot(1/g(x))\) proves the quotient law. In particular, the nonzero-denominator condition is what makes the reciprocal estimate possible.

Worked Examples with the Algebraic Laws

Composition Requires a Direction Check

For a composition \(f(g(x))\), the inner function determines which side of the outer input is approached. This is essential when applying a one-sided limit of \(f\). The inner limit alone is not enough: its direction must agree with the outer limit being used.

To justify the result, take any \(\varepsilon>0\). By the definition of the outer one-sided limit, there is an \(\eta>0\) such that whenever \(t\) is on the specified side of \(b\) and \(0<|t-b|<\eta\), we have \(|f(t)-L|<\varepsilon\). Since \(g(x)\) approaches \(b\) from that side, there is a \(\delta>0\) such that inputs \(x\) sufficiently close to \(a\) on the chosen side give \(g(x)\) on the required side and \(0<|g(x)-b|<\eta\). Taking \(t=g(x)\) gives \(|f(g(x))-L|<\varepsilon\), which proves the composition limit.

The condition \(g(x)\ne b\) near \(a\) ensures that the outer one-sided limit applies to the inputs \(g(x)\). If \(g(x)=b\) for inputs arbitrarily close to \(a\), the outer limit by itself does not tell us the values of \(f(g(x))\) at those inputs. A composition may still have a limit in that situation, but this theorem does not establish it.

Common Mistake: Using the Quotient Law When the Denominator Limit Is Zero

A frequent error is to treat a zero denominator limit as if it were an ordinary number in the quotient law. Consider \(f(x)=x\) and \(g(x)=x\) as \(x\to0^+\). Both component limits are \(0\), but the quotient law does not apply because the denominator limit is zero.

The expression \(0/0\) is not a defined quotient, and the hypotheses of the law are absent. For \(x>0\), \(x/x=1\), so in fact

The correction is not to assign a value to \(0/0\), but to check the quotient law’s condition before using it. When the denominator limit is zero, the law gives no conclusion; the expression must be analyzed by another valid method.

A Reliable Order of Checks

Before combining one-sided limits, check the following:

• Are the component limits taken as \(x\) approaches from the same side?
• Are the component limits finite for the sum, product, or quotient law stated here?
• For a quotient, is the denominator limit nonzero?
• For a composition, does the inner function approach the outer input from the side required by the outer limit?
• Are the expressions defined for inputs sufficiently close to the target from the chosen side?

These checks keep an algebraic calculation tied to the hypotheses that make it valid. The laws are useful because they reduce complicated limits to simpler component limits, but they do not remove the need to establish that those components and operations meet the required conditions.