Ratios of Quadratic Cosine Defects

Why a Ratio of Cosine Defects Has a Finite Limit

When \(x\) is close to zero, both \(1-\cos(ax)\) and \(1-\cos(bx)\) are close to zero. Direct substitution into their ratio therefore gives the indeterminate form \(0/0\). The quadratic cosine defect limit resolves this by describing how quickly each expression approaches zero: the scale in the cosine's input contributes a square.

This squared scaling is important. Replacing \(ax\) by \(x\) changes the leading size of the defect by a factor of \(a^2\), not \(a\). Consequently, the ratio depends on the squared constants, including when either constant is negative.

The Cosine Defect Limit and Scaling

The earlier result on the quadratic cosine defect is \(\displaystyle \lim_{u\to 0}\frac{1-\cos u}{u^2}=\frac12\), with angles in radians. To apply it to a scaled input, let \(u=cx\), where \(c\ne 0\). As \(x\to 0\), \(u\to 0\), and \(u^2=c^2x^2\). Therefore,

This formula says that each cosine defect is approximately a constant times \(x^2\), and that the constant is the square of the input scale divided by \(2\). The following ratio result comes from comparing these two quadratic rates.

Deriving the Ratio Formula

Let \(a,b\ne 0\). For \(x\ne 0\) close to zero, rewrite the desired ratio by dividing the numerator and denominator by \(x^2\):

The numerator of the right-hand side tends to \(a^2/2\), and its denominator tends to \(b^2/2\). Since \(b\ne 0\), the denominator limit \(b^2/2\) is positive, so it is not zero. The quotient law for limits therefore applies. In particular, the original denominator is nonzero for all sufficiently small nonzero \(x\): its quotient by \(x^2\) tends to the positive number \(b^2/2\).

The factor \(1/2\) appears in the quadratic rate for both defects and cancels. The scale factors do not cancel linearly: each rate contains the square of its scale, so their ratio is \(a^2/b^2\).

Worked Examples

A Useful Normalized Form

The same result can be displayed in a form that makes the use of the quadratic cosine defect limit explicit. For nonzero \(x\), multiply and divide by the squared inputs:

As \(x\to 0\), both normalized defect factors tend to \(1/2\), so their ratio tends to \((1/2)/(1/2)=1\). What remains is \(a^2/b^2\). This form also shows why it is important that \(a\) and \(b\) are nonzero: the normalized expressions use \(ax\) and \(bx\) in their denominators, and the stated ratio theorem is for nonzero scales.

The two derivations are equivalent. One compares the quadratic rates after dividing by \(x^2\); the other extracts the squares and compares two normalized defects. In either version, the denominator involved in the quotient of limits has a nonzero limit.

Common Mistake: Forgetting to Square the Scales

A tempting but incorrect calculation is to say that scaling the input by \(a\) makes the defect scale by \(a\), and therefore to conclude that the ratio limit is \(a/b\). For example, this reasoning would give \(3/(-2)=-3/2\) for \(\displaystyle \frac{1-\cos(3x)}{1-\cos(-2x)}\). That answer is wrong: the quadratic rate depends on the square of the input scale.

There is also a useful sign check. For real angles, \(1-\cos(\theta)\ge 0\), and near zero with a nonzero scale the defect is positive for nonzero \(x\) sufficiently close to zero. Thus a negative limit for a ratio of these defects is a warning that a sign or scaling rule has been mishandled.

Conditions and a Reliable Routine

Before applying the formula, check the details of the expression. The constants \(a\) and \(b\) must be fixed real numbers, and both must be nonzero. The limit is taken as \(x\to 0\), and trigonometric arguments are in radians, as required by the quadratic cosine defect limit.

If \(b=0\), the denominator is \(1-\cos(0)=0\) for every \(x\), so this ratio is not defined in a punctured neighborhood of zero. If \(a=0\) and \(b\ne 0\), the numerator is identically zero and the ratio is zero wherever its denominator is nonzero, but that case is outside the nonzero-scale theorem above. Keeping the theorem's hypotheses visible prevents using it where its quotient argument does not apply.

Why the Result Is Useful

The formula provides a direct way to compare two quantities that both vanish quadratically. It avoids trying to substitute into a \(0/0\) expression and makes the rate of vanishing explicit. The result also gives quick consistency checks: swapping numerator and denominator reciprocates the answer, and multiplying either scale by a factor \(k\) multiplies that defect's quadratic rate by \(k^2\).

The most reliable habit is to write the squared constants before simplifying. In particular, a negative input scale still contributes a positive square, and a fractional scale must be squared before it is divided by the denominator's square.