Working Through Trigonometric Limits
Trigonometric limits often become straightforward once an expression is rewritten in a form that matches a standard limit. The main work is choosing that form: a scaled sine quotient, an identity involving cosine, or an inequality that traps the expression between quantities with the same limit.
Throughout this tutorial, angles are measured in radians. The limits used here are as \(u\to 0\): \(\sin u/u\to 1\), \(\tan u/u\to 1\), \(\cos u\to 1\), and \((1-\cos u)/u^2\to 1/2\). These are standard trigonometric limits established in the earlier material on trigonometric limits. When a constant multiple of the variable appears, it is important to account for that scale rather than treating it as if it were one.
A Useful Method for Mixed Expressions
First identify which parts of the expression are causing difficulty. If the expression contains \(\sin(cx)\) or \(\tan(cx)\), create a standard quotient by multiplying and dividing by the argument \(cx\). If it contains \(1-\cos(cx)\), the identity \(1-\cos u=2\sin^2(u/2)\) can turn it into a squared sine quotient. If a bounded trigonometric factor is multiplied by something small, an inequality may be more useful than an identity.
For limits at zero, see whether the expression can be organized around a standard quotient whose argument tends to zero.
Use scaled quotients or a trigonometric identity to expose the factors with known limits.
Apply limit laws only when the component limits exist and any denominator limit is nonzero.
If an expression is trapped between two quantities tending to the same number, the Squeeze Theorem gives its limit.
Worked Example: A Sine Quotient Divided by a Tangent Quotient
Worked Example: A Sine Quotient Divided by a Tangent Quotient
Evaluate \(\displaystyle \lim_{x\to 0}\frac{\sin(4x)}{\tan(3x)}\).
Use \(\tan(3x)=\sin(3x)/\cos(3x)\). For \(x\) sufficiently close to zero, \(\sin(3x)\ne 0\) when \(x\ne 0\), so the expression can be rewritten as
Now insert the scaled quotients. The factors \(4x\) and \(3x\) cancel as shown:
As \(x\to 0\), the first quotient tends to \(1\), the second also tends to \(1\), and \(\cos(3x)\to 1\). Therefore, by the product limit law,
Check: The constants in the scaled arguments give the ratio \(4/3\), and every normalized trigonometric factor tends to \(1\). This confirms the computed value.
Using the One-Minus-Cosine Identity
The standard limit for \(1-\cos u\) has a squared argument in its denominator. If the expression instead contains \(1-\cos(cx)\), the identity \(1-\cos u=2\sin^2(u/2)\) exposes a squared sine quotient. This also provides a direct way to track the constant factors.
Worked Example: A Cosine Difference Over a Sine Factor
Worked Example: A Cosine Difference Over a Sine Factor
Evaluate \(\displaystyle \lim_{x\to 0}\frac{1-\cos(5x)}{x\sin(2x)}\).
Apply the identity with \(u=5x\). Then factor out the arguments needed to form standard sine quotients:
For \(x\ne 0\) close to zero, the denominator quotient is nonzero, and its limit is \(1\). The numerator quotient also tends to \(1\). Thus
Check: The identity contributes the factor \(2\); the squared argument \(5x/2\) contributes \((5/2)^2\), and the denominator \(\sin(2x)\) contributes \(2\). Their combined ratio is \(2(25/4)/2=25/4\), as obtained.
A Complete Squeeze Argument
Standard quotients are not always the best tool. The basic geometric bounds \(|\sin t|\le |t|\) and \(|\cos t|\le 1\), valid for real \(t\), can control expressions that include rapidly oscillating factors. The Squeeze Theorem states that if \(g(x)\le f(x)\le h(x)\) for all \(x\) sufficiently close to a target point, excluding the point if necessary, and both \(g(x)\) and \(h(x)\) tend to \(L\), then \(f(x)\to L\) as well.
Worked Example: An Oscillating Cosine Factor
Worked Example: An Oscillating Cosine Factor
Evaluate \(\displaystyle \lim_{x\to 0}\frac{\sin^2 x}{x}\cos(1/x)\), where the expression is defined for \(x\ne 0\).
The cosine factor need not approach a limit as \(x\to 0\). Instead, bound its absolute value by \(1\), and use \(|\sin x|\le |x|\):
For \(x\ne0\), the absolute-value bound implies
As \(x\to0\), both \(-|x|\) and \(|x|\) tend to \(0\). The Squeeze Theorem therefore gives
Check: The bound holds on both sides of zero, and both bounding functions have the same limit, \(0\). No limit for \(\cos(1/x)\) is needed.
Worked Example: A Ratio of Squared Sines
Worked Example: A Ratio of Squared Sines
Evaluate \(\displaystyle \lim_{x\to 0}\frac{1-\cos(3x)}{\sin^2(2x)}\).
Rewrite the numerator using \(1-\cos u=2\sin^2(u/2)\), then normalize both sine factors:
Each normalized quotient tends to \(1\). The denominator quotient has a nonzero limit, so the quotient limit law applies:
Check: The numerator contributes \(2(3/2)^2=9/2\), and the denominator contributes \(2^2=4\). Their ratio is \((9/2)/4=9/8\), agreeing with the result.
Common Mistake: Ignoring the Scale
A frequent error is to reason that because both \(\sin(4x)\) and \(\sin(7x)\) tend to zero, their quotient must tend to \(1\). The fact that numerator and denominator both tend to zero does not give the quotient's limit. Their rates of approach matter.
The correction is to expose the scale factors:
The same care applies to identities and inequalities. Keep track of constants when normalizing arguments, check that a denominator is nonzero near the limit point, and verify that squeeze bounds hold throughout a punctured neighborhood rather than only at selected values.
Choosing a Method
| Expression feature | Useful approach |
|---|---|
| \(\sin(cx)\) or \(\tan(cx)\) near zero | Form a scaled quotient and retain the factor \(c\). |
| \(1-\cos(cx)\) | Use \(1-\cos u=2\sin^2(u/2)\) or the standard quadratic cosine limit. |
| A bounded trigonometric factor times a small factor | Use absolute-value bounds and the Squeeze Theorem. |
| A quotient of expressions with known limits | Check that the denominator limit is nonzero before applying the quotient law. |
These methods matter because they turn unfamiliar-looking expressions into a small set of reusable patterns. A reliable solution does not guess the limit from a few numerical values: it rewrites the expression, states why the limit laws or squeeze bounds apply, and checks the constants created by the arguments.
Check Your Understanding
For each problem, identify the useful rewrite or bound before calculating.
- Evaluate \(\displaystyle \lim_{x\to0}\frac{\sin(6x)}{\tan(2x)}\).
- Evaluate \(\displaystyle \lim_{x\to0}\frac{1-\cos(4x)}{x\sin(3x)}\).
- Use an inequality to evaluate \(\displaystyle \lim_{x\to0}x\sin(1/x)\).
- Explain why \(\sin(5x)/\sin(2x)\) does not have limit \(1\), and calculate its limit.
- In a quotient limit calculation, what condition on the denominator's limit is needed to use the quotient law?
