Sandwich Theorem Proof

To prove the sandwich theorem we can use the epsilon-delta definition of limit, which is called algebraic proof. The following heading shows detailed proof of the sandwich theorem using limits.

Algebraic Proof Using Definition of Limit

Sandwich theorem can be easily proved using the definition of limit. Assume three real-valued functions g(x), f(x), and h(x) such that g(x) ≤ f(x) ≤ h(x) and lim_x→a g(x) = lim_x→a h(x) = L. 

Then by the definition of limits,

lim_x→a g(x) = L signifies ∀ ∈ > 0, ∃ δ₁ > 0 such that |x – a| < δ₁ ⇒ |g(x) – L| < ∈

|x – a| < δ₁ ⇒ -∈ < g(x) – L < ∈   … (i)

lim_x→a h(x) = L signifies ∀ ∈ > 0, ∃ δ₂ > 0 such that |x – a| < δ₂ ⇒ |h(x) – L| < ∈

|x – a| < δ₂ ⇒ -∈ < h(x) – L < ∈    … (ii)

Given, g(x) ≤ f(x) ≤ h(x)

Subtracting L from each side of the inequality

g(x) – L ≤ f(x) – L ≤ h(x) – L

Taking δ = minimum {δ₁, δ₂}, Nowr |x – a| < δ,

-∈ < g(x) – L ≤ f(x) – L ≤ h(x) – L < ∈                 [using (i) and (ii)]

-∈ < f(x) – L < ∈

lim_x→a f(x) = L

Thus, this proved the Sandwich Theorem.

Geometric Proof of cos x < sin x/x < 1

We can understand the geometric proof of the statement cos x < sin x/x < 1, as follows

To Prove:  cos x < sin x/x < 1, for 0 < |x| < π/2

Also, using trigonometric identites

• sin (– x) = – sin x
• cos( – x) = cos x

We are now proving the inequality in eq(i)

Draw a unit circle with centre O

Now if ∠COA is x radians and 0 < x < π/2

For the figure,

Area of ΔOAC < Area of sector OAC < Area of Δ OAB

1/2×OA×CD < x/2π × π(OA)² < 1/2×OA×AB

Cancelling OA from each side

CD < x.OA < AB…(i)

In ΔOCD

sin x = CD/OC = CD/OA

CD = OA sin x…(ii)

Similarly, In ΔOAB

tan x = AB/OA

AB = OA tan x…(iii)

Now, from eq (i), (ii) and (iii) we get

sin x < x < tan x  [given 0<x<π/2]

Dividing sin x from each side

sin x/ sin x < x/(sin x) < tan x/ (sin x)

1 < x/(sin x) < 1/ (cos x)

Taking reciprocal,

cos x < (sin x)/(x) < 1

Thus, Proved.

Sandwich Theorem also called Sandwich Rule or Squeeze Theorem, is an important theorem in calculus involving limits and it is used to find the limit of some functions when the normal methods of finding the limit fail. Suppose we have to find the limit such that lim_x->a f(x) gives an indeterminant form and solving the limit algebraically does not help then we find the limit of f(x) at a^– and a⁺ and if both the limits are equal then this limit also becomes the limit of f(x) at a. This is the sandwich theorem or squeeze theorem. 

The sandwich theorem is typically applied to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. Famous Greek mathematicians Archimedes was the first to use this concept. In this article, we will learn about Sandwich Theorem or Squeeze Theorem statement, its proof, examples, and others in detail.