The Squeeze Theorem

The Squeeze Theorem:

(a) Let f, g, and h be functions defined on an interval (a,b) with a < c < b, and suppose that

g(x) ≤ f (x) ≤ h(x)

for all x in the interval (a,b). Then

lim g(x) ≤ lim f (x) ≤ lim h(x) x→c x→c x→c

(b) Let f, g, and h be functions defined on an interval (a,∞) and suppose that

g(x) ≤ f (x) ≤ h(x)

for all x in the interval (a,∞). Then

lim g(x) ≤ lim f(x) ≤ lim h(x) x→ ∞ x→ ∞ x→ ∞

A similar statement is true as x →-∞.

Return to the Limits at Infinity page.