Why Cancelling a Factor Preserves a Limit

Why a Hole Does Not Determine a Limit

A limit describes the values of a function near a target, not necessarily the value of the function at the target itself. This distinction explains why cancelling a shared factor can reveal a limit even when the original expression is undefined at the point we are approaching.

For example, the expression \(\frac{x^2-4}{x-2}\) is undefined at \(x=2\), because its denominator is zero there. But for nearby values \(x\ne 2\), its numerator and denominator have a shared factor. Cancelling that factor gives an expression whose values are easy to examine. The cancellation does not make the original expression defined at \(x=2\); it shows that the two expressions agree at all sufficiently nearby points other than \(2\).

What It Means to Agree Near a Target

Suppose \(f\) and \(g\) are functions defined for inputs near \(a\), perhaps with either function undefined at \(a\). They agree on a punctured neighborhood of \(a\) if there is some positive distance around \(a\) such that \(f(x)=g(x)\) whenever \(x\) is within that distance but \(x\ne a\).

The phrase “but \(x\ne a\)” matters. A limit considers values approaching \(a\), not the value at \(a\). Thus, two functions can have the same limit even if one or both are undefined at \(a\), or if their values at \(a\) differ.

This theorem is the reason cancellation preserves a limit. Cancelling a factor produces an expression equal to the original one at every nearby input where the cancellation is valid. The limit can then be found using the simpler expression.

Why the Theorem Works

Here is a justification using the definition of a limit. The statement \(\lim_{x\to a}g(x)=L\) means that for every positive tolerance \(\varepsilon\), there is a positive distance \(\delta\) such that whenever \(0<|x-a|<\delta\), the value \(g(x)\) is within \(\varepsilon\) of \(L\).

Choose \(\delta\) small enough that \(f(x)=g(x)\) whenever \(0<|x-a|<\delta\). Since \(g\) tends to \(L\), for any given \(\varepsilon>0\) we can also choose a distance that ensures \(|g(x)-L|<\varepsilon\). Use the smaller of these distances. Then \(f(x)=g(x)\), so \(f(x)\) is also within \(\varepsilon\) of \(L\). This proves that \(f\) tends to \(L\).

In an algebraic limit problem, the equality \(f(x)=g(x)\) usually comes from cancelling a factor. The restriction \(x\ne a\) is essential: if the shared factor is zero at \(a\), division by it is not valid at \(a\). Limits use the nearby equality, not an invalid substitution at the target.

Worked Examples: Cancel and Evaluate

Conditions and Limits of Cancellation

Cancellation is an equality of expressions only where the cancelled factor is nonzero. If the factor vanishes at the target, that usually still leaves a whole punctured neighborhood where cancellation is valid. For example, \(x-a\ne 0\) whenever \(x\ne a\). The limit theorem then transfers the limit from the simplified expression to the original one.

After cancellation, check what remains. If the simplified expression is a polynomial, its limit can be found by evaluation. If it is a quotient, its denominator must be nonzero at the target before direct evaluation using the quotient limit law. Cancellation by itself does not guarantee a finite limit or even settle the limit.

A Common Mistake: Cancelling at the Target

A tempting but incorrect argument is to substitute the target first and treat the shared zero factor as if it can be cancelled there. For instance, someone might write

This is not a valid evaluation: the first quotient has denominator \(0\), and the cancellation step divides by \(2-2=0\). The correction is to state the restriction before cancelling:

Then use the equality for nearby inputs other than \(2\) to evaluate the limit. The answer \(4\) is the limit, not the value of the original quotient at \(2\).

Another related error is cancelling part of a sum rather than a factor. Cancellation applies to common factors of a numerator and denominator, not to terms that merely appear in both. Always factor the expressions and check the equality after cancelling.

Why Removable Holes Matter

A removable hole occurs when an expression is undefined at a point but has a finite limit there. The cancelled expression describes the nearby values and supplies the limit; it does not automatically fill in the missing function value. For example, \(\frac{x^2-4}{x-2}\) has a limit of \(4\) at \(x=2\), although the original quotient has no value there. A new function could be defined to equal \(4\) at \(2\), but that would be a separate definition.

This distinction is useful throughout limit work: first establish where two formulas agree, then use the simpler formula to study nearby behavior, and finally keep the limit separate from any value assigned at the target.