What are some examples of the squeeze theorem?

Examples of the squeeze theorem, g(x) ? f(x) ? h(x), show that if f(x) is always greater than g(x) and if f(x) is always less than h(x), then when g(x) is equal to h(x), f(x) must also be equal. Since f(x) always squeezes between g(x) and h(x), it must be equal when g(x) and h(x) are equal.

Here are two examples of the Squeeze theorem:

Peter always works the same or more hours than John. Peter always works the same or fewer hours than Cindy. John worked 10 hours yesterday. Cindy worked 10 hours yesterday. How many hours did Peter work yesterday?

g(x) = hours John works

h(x) = hours Cindy works

f(x) = hours Peter works.

Where a = yesterday,

lim g(x)= 10 hours

x?a

lim h(x) = 10 hours

x?a

Therefore,

lim f(x) = 10 hours

x?a

Peter worked 10 hours yesterday.

Trisha always runs at the same or a faster pace than Sheila. Trisha always runs at the same or a slower pace than Jill. Today, Sheila ran 5 miles in 45 minutes and Jill ran 5 miles in 45 minutes. Today Trisha also ran for 45 minutes. How far did she run?

g(x) = distance Sheila runs in 45 minutes

h(x) = distance Jill runs in 45 minutes

f(x) = distance Trisha runs in 45 minutes

Where a= today,

lim g(x)= 5 miles

x?a

lim h(x) = J5 miles

x?a

Therefore,

lim f(x) = 5 miles

x?a

Trisha ran 5 miles in 45 minutes today.