When Oscillation Is Controlled by a Shrinking Power
A function can oscillate more and more rapidly as its input approaches zero without settling toward a single value. For example, \(\sin(1/x)\) keeps oscillating between \(-1\) and \(1\) as \(x\) approaches zero through nonzero values.
Multiplying such a function by a positive power of \(x\) can change the limit completely. The oscillation may persist, but the power becomes small enough to force the whole product toward zero. The key is not to find a limit for the oscillating factor. Instead, we bound it.
What “Bounded Near Zero” Means
A function \(g\) is bounded near zero if there are constants \(M\ge 0\) and \(\delta_0>0\) such that \(|g(x)|\le M\) whenever \(0<|x|<\delta_0\) and \(g(x)\) is defined. The bound must work throughout some punctured neighborhood of zero; it is not enough for \(g\) to be bounded at only selected input values.
For an oscillating factor such as \(\sin(1/x)\), the familiar inequality \(-1\le \sin(t)\le 1\), valid for every real \(t\), gives \(|\sin(1/x)|\le 1\) for every nonzero \(x\). The frequency of the oscillation does not affect this bound.
The absolute value is useful because it handles either sign of the product and either side of zero. The factor \(|x|^n\) tends to zero for every positive integer \(n\), so both bounds in the theorem tend to zero.
Why the Bound Proves the Limit
Here is a direct epsilon proof. It also shows exactly how small \(x\) needs to be. Suppose first that \(M>0\), and let \(\varepsilon>0\) be given. Choose \[ \delta=\min\left(\delta_0,\left(\frac{\varepsilon}{M}\right)^{1/n}\right). \] Whenever \(0<|x|<\delta\), the local bound on \(g\) applies, and \(|x|<(\varepsilon/M)^{1/n}\). Therefore \[ |x^n g(x)|=|x|^n|g(x)|\le M|x|^n<\varepsilon. \] This is the definition of \(\lim_{x\to0}x^n g(x)=0\). If \(M=0\), then \(g(x)=0\) throughout the punctured neighborhood, so the product is already zero there and the limit is zero.
The same reasoning can be read as an application of the Squeeze Theorem: the product lies between \(-M|x|^n\) and \(M|x|^n\), and each bound tends to zero. The important insight is that the size of the product is controlled by its absolute value, even when its sign keeps changing.
Worked Examples
Worked Example: A Squared Power Times a Sine
Find \(\displaystyle\lim_{x\to0}x^2\sin(1/x)\).
For \(x\ne0\), the sine bound gives \[ -1\le\sin(1/x)\le1. \] Since \(x^2\ge0\), multiplying the inequalities by \(x^2\) preserves their order: \[ -x^2\le x^2\sin(1/x)\le x^2. \] As \(x\to0\), both \(-x^2\) and \(x^2\) tend to zero. The Squeeze Theorem therefore gives \[ \lim_{x\to0}x^2\sin(1/x)=0. \]
Check: the absolute value of the expression is at most \(x^2\), and \(x^2\) can be made smaller than any positive tolerance by taking \(x\) sufficiently close to zero. This confirms that the product tends to zero even though the sine factor does not settle to a limit.
Worked Example: A Bounded Factor with a Larger Bound
Find \(\displaystyle\lim_{x\to0}x^3\bigl(2+\cos(1/x^2)\bigr)\).
Because \(-1\le\cos(1/x^2)\le1\), adding \(2\) gives \[ 1\le2+\cos(1/x^2)\le3. \] In particular, \(|2+\cos(1/x^2)|\le3\). Taking absolute values of the product, \[ \left|x^3\bigl(2+\cos(1/x^2)\bigr)\right| =|x|^3\left|2+\cos(1/x^2)\right| \le3|x|^3. \] As \(x\to0\), \(3|x|^3\to0\). Thus the product’s absolute value tends to zero, which means the product itself tends to zero: \[ \lim_{x\to0}x^3\bigl(2+\cos(1/x^2)\bigr)=0. \]
Check: the factor \(2+\cos(1/x^2)\) never exceeds \(3\) in absolute value, so the whole expression is no larger in absolute value than \(3|x|^3\). This upper bound tends to zero from both sides, as required.
Worked Example: Adding Two Oscillating Terms
Find \(\displaystyle\lim_{x\to0}x^4\bigl(\sin(1/x)+\cos(1/x)\bigr)\).
The triangle inequality and the sine and cosine bounds give \[ |\sin(1/x)+\cos(1/x)| \le|\sin(1/x)|+|\cos(1/x)| \le1+1=2. \] Consequently, \[ \left|x^4\bigl(\sin(1/x)+\cos(1/x)\bigr)\right| \le2|x|^4. \] Since \(2|x|^4\to0\) as \(x\to0\), the squeeze argument yields \[ \lim_{x\to0}x^4\bigl(\sin(1/x)+\cos(1/x)\bigr)=0. \]
Check: the sum of oscillating terms is bounded in absolute value by \(2\), so its product with \(x^4\) is bounded in absolute value by \(2|x|^4\). That bound tends to zero, verifying the claimed limit.
Positive Real Powers on the Right of Zero
The same idea works for a non-integer positive power when the domain is restricted to positive \(x\). If \(p>0\), then \(x^p\to0\) as \(x\to0^+\). Thus, if \(|g(x)|\le M\) for all sufficiently small positive \(x\), then \[ |x^p g(x)|\le Mx^p\longrightarrow0. \] This proves the corresponding right-hand limit. The restriction matters: a general real power \(x^p\) need not be real-valued for negative \(x\). For a two-sided limit with an integer power, the earlier theorem applies directly.
Worked Example: A Fractional Power on a Positive Domain
Find \(\displaystyle\lim_{x\to0^+}\sqrt{x}\cos(1/x)\).
For \(x>0\), the cosine bound gives \(|\cos(1/x)|\le1\), so \[ \left|\sqrt{x}\cos(1/x)\right| =\sqrt{x}\,|\cos(1/x)| \le\sqrt{x}. \] As \(x\to0^+\), \(\sqrt{x}\to0\). Hence \[ \lim_{x\to0^+}\sqrt{x}\cos(1/x)=0. \]
Check: the absolute value of the product is no greater than \(\sqrt{x}\), and this bound tends to zero on the stated positive domain. The one-sided conclusion is therefore verified.
A Common Mistake: Requiring the Oscillating Factor to Have a Limit
A tempting but incorrect argument is: “\(\sin(1/x)\) has no limit at zero, so \(x^2\sin(1/x)\) has no limit either.” The conclusion does not follow. A product can have a limit even when one factor does not; the shrinking power may control the product’s size.
For a concrete correction, the incorrect calculation would stop after observing that \(\sin(1/x)\) oscillates and declare the product’s limit nonexistent. Instead, use the bound: \[ \left|x^2\sin(1/x)\right| \le x^2. \] Since \(x^2\to0\), the product tends to zero. The mistake is treating the factors as if their limiting behavior could be considered independently, even though the oscillating factor has no limit to substitute. The correct method bounds the entire product.
A related caution is that boundedness is an essential condition in this result. The argument does not say that every function multiplied by \(x^n\) produces a product tending to zero. It applies when the other factor has one fixed finite bound throughout a punctured neighborhood. If that factor is unbounded, this particular estimate cannot establish the limit, and the product must be analyzed using an appropriate method.
How to Use the Method
Write the expression as a power that tends to zero multiplied by a function whose size can be bounded.
Use a known inequality or the triangle inequality to obtain \(|g(x)|\le M\) for a fixed constant \(M\).
Use \(|x^n g(x)|\le M|x|^n\), or the corresponding right-sided bound for a positive real power.
If the upper bound tends to zero, the product tends to zero by the squeeze argument.
Why This Pattern Matters
Oscillations occur in many limit problems, and their frequency may grow without bound as the input approaches a point. A direct attempt to evaluate the oscillating factor can therefore fail: it may not have a limit at all. The bounding method asks a more useful question: how large can the entire expression be?
If a positive power tends to zero and the other factor remains bounded, the answer is controlled by a quantity that tends to zero. This is why the power can “overwhelm” the oscillation. The precise tool is an inequality, not a prediction about which values the oscillating factor will take near zero.
Check Your Understanding
For each question, identify the bound that controls the product and state whether the limit is two-sided or one-sided.
- Why does boundedness of \(g\) near zero give a bound on \(|x^n g(x)|\)?
- Find \(\displaystyle\lim_{x\to0}x^2\cos(1/x)\) by stating a suitable inequality.
- If \(|h(x)|\le4\) near zero, what conclusion follows for \(\displaystyle\lim_{x\to0}x^5h(x)\)?
- Why does \(\sin(1/x)\) having no limit not prevent \(x^3\sin(1/x)\) from having a limit?
- For \(p>0\), what domain restriction is used when applying the method to \(x^p\) as \(x\) approaches zero?
