Understanding the Squeeze Theorem and Its Applications in Limits

An Introduction to the Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is one of the most important tools for calculating the limit of functions whose behavior is not directly recognizable.
This theorem is useful when a function is trapped between two other functions that share the same limit.

The Formal Statement of the Squeeze Theorem

If we have three functions such as f(x), g(x), and h(x) such that:

f(x) ≤ g(x) ≤ h(x)

and if:

lim (x → a) f(x) = lim (x → a) h(x) = L

then according to the Squeeze Theorem we have:

lim (x → a) g(x) = L

This theorem is especially useful when the middle function cannot be evaluated directly.

Why Is the Squeeze Theorem Important?

This theorem is particularly valuable in the following situations:

• When the function is oscillatory
• When the function is undefined at the point of interest
• When direct computation of the limit is difficult or impossible
• When analyzing the behavior of trigonometric functions near zero

A Classic Example of the Squeeze Theorem

One of the most famous examples involves the function:

g(x) = sin(x) / x

For x near zero, we know:

-1 ≤ sin(x)/x ≤ 1

But with more precise analysis, we can show:

cos(x) ≤ sin(x)/x ≤ 1

And since:

lim (x → 0) cos(x) = 1

we conclude using the Squeeze Theorem:

lim (x → 0) sin(x)/x = 1

Another Example of the Squeeze Theorem

Consider the function:

g(x) = x² sin(1/x)

We know that:

-1 ≤ sin(1/x) ≤ 1

Therefore:

-x² ≤ x² sin(1/x) ≤ x²

And since:

lim (x → 0) x² = 0

we conclude:

lim (x → 0) x² sin(1/x) = 0

Conclusion

The Squeeze Theorem is one of the most powerful tools for calculating the limit of complex functions.
By placing a function between two others with known limits, it allows us to determine the limit even in challenging situations.
Understanding this theorem is essential for success in advanced topics such as continuity, derivative, and mathematical analysis.