1.8 Determining Limits Using the Squeeze Theorem

What the Squeeze Theorem says

If g(x) ≤ f(x) ≤ h(x) for all x near a value c (except possibly at c), and lim g(x) = lim h(x) = L as x→c, then lim f(x) = L too. The function f is 'squeezed' between two functions that converge to the same limit, forcing f to converge there as well.

Classic use: trig limits

The Squeeze Theorem proves lim(x→0) sin(x)/x = 1 and lim(x→0) x·sin(1/x) = 0. For the second, since -1 ≤ sin(1/x) ≤ 1, multiplying by x gives -|x| ≤ x·sin(1/x) ≤ |x|. As x→0, both bounds → 0, so the middle function's limit is 0.

When to use it

Use the Squeeze Theorem when a function is too complicated or oscillatory to evaluate directly (like something times sin or cos of 1/x), but you can bound it between two simpler functions with a known, equal limit at the point of interest. Always confirm the inequality holds near c, and that both bounding limits truly match.

Key terms

Squeeze Theorem (statement)

If g(x) ≤ f(x) ≤ h(x) near c and lim g(x) = lim h(x) = L as x→c, then lim f(x) = L.

Key trig limit proved by squeezing

lim(x→0) sin(x)/x = 1

lim(x→0) x·sin(1/x)

0, since -|x| ≤ x·sin(1/x) ≤ |x| and both bounds → 0.

When to use the Squeeze Theorem

When a function is oscillatory or too complex to evaluate directly, but can be bounded between two simpler functions with equal limits.

Bounding inequality for cos and sin

-1 ≤ sin(x) ≤ 1 and -1 ≤ cos(x) ≤ 1 for all real x — commonly used to set up squeeze inequalities.

Try an AP-style question

If 1 − x² ≤ f(x) ≤ 1 + x² for all x near 0, what is lim(x→0) f(x)?

1. 2
2. 1
3. 0
4. Cannot be determined