MATH1851 Chap.2 Sandwich Theorem and Oscillatory Limits
Sandwich Theorem and Oscillatory Limits
Bound the difficult function
The Sandwich Theorem is a comparison argument. If a target function remains between a lower and upper function near the point, and both outer functions approach the same finite value, the middle function must approach that value too.
The inequality and the equal outer limits are separate hypotheses, so a complete proof displays both before invoking the theorem.
Oscillation can survive while amplitude disappears
The lecture contrasts an undamped sine of a reciprocal, whose values keep ranging between minus one and one near zero, with products that multiply the oscillation by a factor tending to zero. Frequency does not need to settle.
The shrinking envelope is enough. Absolute values provide a clean two-sided bound and prevent an inequality from reversing when the variable is negative.
Know when the theorem is silent
A bounded function is not automatically squeezed. If the proposed outer limits differ, the corridor remains open and the theorem gives no conclusion. If the bounds are unbounded, they do not force a finite result.
A failed Sandwich argument does not prove the original limit fails; it only shows that more information or another method is needed.
Reduce related trigonometric limits
The fundamental trigonometric limit supplies a reusable standard form. Rewrite the inner variable so the numerator contains the sine of that variable and the denominator contains the same variable.
Coefficients become ordinary multiplicative factors. The transformation is complete only after you verify that the new inner variable tends to zero.
Audit the squeeze
Before citing the theorem, check that the ordering holds throughout a punctured interval and not only at sampled inputs. Then display both outside limits on the same line.
The shared value is what forces the middle function, while the inequality identifies which function is actually trapped.
What this chapter covers
- 01
The lower and upper inequality
- 02
Equal outer limits as the forcing condition
- 03
Oscillation with a shrinking envelope
- 04
Absolute-value bounding across both sides
- 05
Fundamental trigonometric limit reductions
- 06
Unequal and unbounded bounds as failure cases
Worked example · free
Trap an oscillating product
- 2Bound the trigonometric factor in magnitude.
- 3Construct lower and upper functions valid on both sides.
- 2Compute the shared outer limit and invoke the theorem.
Key terms
- Lower Bound
- A function whose values remain at or below the target function on the relevant nearby interval.
- Upper Bound
- A function whose values remain at or above the target function on the relevant nearby interval.
- Shared Limit
- The same finite value reached by both bounding functions, which forces the middle limit.
- Oscillation
- Repeated variation that prevents outputs from settling when its amplitude does not vanish.
- Vanishing Envelope
- A magnitude bound tending to zero even while the bounded middle factor continues to oscillate.
- Trig Limit
- The standard limit of sine theta divided by theta as theta approaches zero, used to reduce related forms.
Sandwich Theorem and Oscillatory Limits FAQ
Does oscillation always mean a limit fails?
No. Fixed-amplitude oscillation prevents settling, but a vanishing multiplier can force the whole expression toward one value. Bound the absolute magnitude. If that bound tends to zero, the Sandwich Theorem can establish a zero limit despite infinitely rapid oscillation.
What must I write to use the Sandwich Theorem?
Write the lower-to-middle-to-upper inequality on a nearby punctured interval, calculate the lower and upper limits, show those values are equal and then state the theorem's conclusion. A picture or a claim that the function is bounded does not replace those steps.
If the bounding limits differ, does the target limit fail?
Not necessarily. Unequal outer limits mean only that these bounds do not determine the answer. The middle function might still have a limit, or it might not. You need tighter bounds or a different method before making a verdict about existence.
Exam move
Build bounds from familiar ranges such as absolute sine and cosine being at most one. Practise writing the full inequality before computing any limit. For every failed attempt, classify the reason: ordering not established, bounds valid only on one side, outer limits unequal or outer functions unbounded. Then practise trigonometric reductions by naming a temporary inner variable and checking that it tends to zero.
Your final line should identify the theorem and the shared bound value, not merely report the answer.
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