Question

how to solve sandwich theorem​

Answer 1

Answer:

The sandwich theorem is also known as Squeeze theorem or Pinch theorem.

Step-by-step explanation:

In the above figure, we see Δ ABE, Δ ADF, Δ ADB and sector ADB.

Now, AB = AD (isosceles traingle)

Area (ΔABD) < Area (sector ADB) < Area (ΔADF)

1/2.AD.EB < x/2π.π.AD2 < 1/2.AD.DF

Cancelling the common terms from all sides, we get

EB < x.AD < DF

From Δ ABE, sinA = EB/AB, so EB = AB sinx (angle A= angle X)

Also, tanA = DF/AD, so DF = ADtanX

But AB = AD and

tanA = sinX/CosX

so, AD.sinA < x.AD < AD.sinA / cosA

= 1 < x / sinX < 1 / cosX

Taking reciprocals we get,

cosx < sinx / x < 1

Hence proved.

Using the above theorem we can easily prove some other trigonometric identities such as

limx->0sinx/x =1

limx->0 (1 – cosx)/x =0

We have to keep in mind some points while evaluating the limits.Suppose while evaluating a function where limx->af(x)/g(x) exists such that f(x) = 0 and G(x) = 0. Then in such cases, we rewrite f(x) and G(x) in such ways that we get 2 functions. This can be said as f(x) = f'(x).f”(x) such that f'(x) = 0.

Similarly we get g(x)=g'(x).g”(x) such that g'(x)=0. We then cancel out the common terms and we get the new limits as

limx->af(x) / g(x) = f'(a)/g'(a)