One-Sided Limits At Radical Endpoints

Introduction

A square root restricts the inputs of a real-valued function: its radicand must be nonnegative. When the radicand becomes zero at an endpoint, the function may be defined on only one side of that endpoint. A one-sided limit must respect this domain restriction.

The domain tells us which direction to approach from. After that, the sign and size of the expression determine whether the limit is finite, infinite, or does not exist. In particular, a denominator approaching zero can produce either positive or negative infinity, depending on its sign on the allowed side.

Which Side of the Endpoint Is Allowed?

For a real square root \(\sqrt{g(x)}\), the radicand must satisfy \(g(x)\ge 0\). If \(g(x)=x-a\), this condition is \(x\ge a\), so the domain begins at \(a\) and points to the right. If \(g(x)=a-x\), the condition is \(x\le a\), so the domain ends at \(a\) and points to the left.

For example, \(f(x)=\sqrt{x-2}\) is real-valued only for \(x\ge 2\), so \(\lim_{x\to2^+}f(x)\) is meaningful, but there are no domain inputs immediately to the left of \(2\) on which to form \(\lim_{x\to2^-}f(x)\). The missing left-hand limit is not a limit equal to zero or infinity; the function simply has no values on that side near the endpoint.

When an affine radicand is written as \(c(x-a)\) with \(c>0\), its nonnegativity also requires \(x\ge a\). If it is \(c(a-x)\) with \(c>0\), it requires \(x\le a\). For more complicated radicands, determine the sign near the endpoint before choosing a side.

Roots Approach Zero, but Their Reciprocals Can Grow Without Bound

If \(t\ge0\) and \(t\to0\), then \(\sqrt{t}\to0\). This follows directly from the definition of a limit: for any \(\varepsilon>0\), if \(0\le t<\varepsilon^2\), then \(0\le\sqrt{t}<\varepsilon\). The sign restriction matters. A real square root near a zero radicand is nonnegative.

By contrast, when \(t\to0^+\), the reciprocal \(1/\sqrt{t}\) grows without bound. To see this, take any \(M>0\). If \(0<t<1/M^2\), then \(\sqrt{t}<1/M\), and hence \(1/\sqrt{t}>M\). Thus the reciprocal tends to \(+\infty\).

This simple comparison is useful, but not every quotient at an endpoint has this form. If both numerator and denominator approach zero, the result cannot be read off from those two facts alone. Algebra, often using a conjugate, can reveal the behavior.

Worked Example: A Positive Infinite Limit at a Right Endpoint

Track the Denominator’s Sign

A denominator tending to zero does not determine the sign of an infinite limit by itself. Check whether it approaches zero through positive or negative values on the allowed side. The numerator matters too: if it stays positive, the quotient has the denominator’s sign; if it stays negative, the quotient has the opposite sign.

The same sign check applies when a square root is only part of a more complicated denominator. Use the domain to determine which side is available, and then use the factors or a sign chart to establish whether the denominator is positive or negative there.

When Both Numerator and Denominator Approach Zero

If both parts of a quotient approach zero, direct substitution gives the indeterminate form \(0/0\). This does not specify the limit. For radical expressions, multiplying by the conjugate often exposes a factor that can be canceled. The conjugate identity works because \((u-v)(u+v)=u^2-v^2\).

A Practical Endpoint Procedure

For a one-sided limit involving a square root at a domain endpoint, keep the domain and the limit calculation separate at first. The domain decides which inputs can approach the endpoint; the algebra and signs decide what the function values do.

Common Mistake: Ignoring the Allowed Side’s Sign

In the left-endpoint example, a common error is to treat \(x-4\) as positive because it is a denominator close to zero. That gives an incorrect positive infinite limit. But for every \(x<4\), \(x-4<0\). The numerator \(\sqrt{4-x}\) is positive, so the quotient is negative.

The correction is to use \(x-4=-(4-x)\), which gives \(-1/\sqrt{4-x}\) and therefore the limit \(-\infty\). The sign is not a detail to check after the calculation: it is part of the limit analysis.

Why the Domain Restriction Matters

At a radical endpoint, domain information determines whether a one-sided limit is available at all. For example, \(\sqrt{x-a}\) has real values only for \(x\ge a\); a left-hand approach through the real domain is impossible. If a denominator also approaches zero, its sign on the permitted side determines whether an unbounded quotient tends to \(+\infty\) or \(-\infty\). If both numerator and denominator vanish, a conjugate can reveal a finite limit or a different behavior.

These checks prevent two distinct errors: claiming a limit from a side where the function is not defined, and assigning an infinite limit without verifying its sign. They also prepare you to analyze the behavior of functions near points where values grow without bound.