The Squeeze Theorem (also known as the Sandwich Theorem or Pinching Theorem) is used to find the limit of a function when the function itself is difficult to evaluate directly, but it is "squeezed" between two simpler functions whose limits are easier to determine. The theorem works when these simpler functions converge to the same limit.

Steps to know when to use the Squeeze Theorem:

1. Identify a challenging limit: If you're trying to evaluate $\lim_{x \to a} f(x)$ and the function $f(x)$ is difficult to compute directly.

2. Find two bounding functions: Look for two functions, say $g(x)$ and $h(x)$, such that: $g(x) \leq f(x) \leq h(x)$ for all $x$ near $a$ (but not necessarily at $a$).

3. Check the limits of the bounding functions: Compute the limits of $g(x)$ and $h(x)$ as $x \to a$: $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$

4. Apply the theorem: If both $g(x)$ and $h(x)$ converge to the same limit $L$, then by the Squeeze Theorem: $\lim_{x \to a} f(x) = L$

When to Use the Squeeze Theorem:

• When you know or suspect that $f(x)$ lies between two functions for which the limits are easier to calculate.
• When dealing with functions that oscillate, like trigonometric functions (e.g., $\sin x$ or $\cos x$) multiplied by terms that approach 0.
• When $f(x)$ involves absolute values or other expressions that naturally lend themselves to inequalities.

Example:

A classic example is finding the limit of: $\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right)$ We know that $-1 \leq \sin\left(\frac{1}{x}\right) \leq 1$, so: $-x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2$ Since $\lim_{x \to 0} -x^2 = \lim_{x \to 0} x^2 = 0$, by the Squeeze Theorem: $\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0$

Would you like a more detailed example, or do you have any specific questions?

Relative Questions:

1. Can the Squeeze Theorem be applied to functions with infinite limits?
2. How do you find the two bounding functions when applying the Squeeze Theorem?
3. Are there specific types of functions for which the Squeeze Theorem is most effective?
4. Can the Squeeze Theorem be used for limits at infinity?
5. What are some common mistakes students make when applying the Squeeze Theorem?

Tip:

Always ensure that the two bounding functions have the same limit before concluding the limit of the squeezed function.