What is the value of ?

Concept:

The Squeeze Theorem (The Sandwich Theorem): is used on a function where it will be almost impossible to differentiate.

• The squeeze theorem states that if we define functions such that h(x) ≤ f(x) ≤ g(x) and if , then .

Calculation:

We know that -1 ≤ sin θ ≤ 1.

⇒ -1 ≤  ≤ 1

Since, e^x is a strictly increasing function for all real values of x, we can say that:

⇒ e^-1 ≤  ≤ e¹

Also, since x² ≥ 0, we can say that:

⇒ x²e-1 ≤  ≤ x²e1

⇒ 

So, we can consider h(x) = , f(x) =  and g(x) = x²e.

Now, .

And .

Since, , we must have .

Hence, .