#Study Notes: Squeeze Theorem and Trigonometric Limits

#Table of Contents

1. Squeeze Theorem
   • Definition
   • Formal Statement
   • Worked Example
2. Trigonometric Limits
   • Important Trigonometric Limit Theorems
   • Worked Examples
3. Practice Questions
4. Glossary
5. Summary and Key Takeaways

#Squeeze Theorem

#Definition

The squeeze theorem is a method used to determine the limit of a function that is bounded above and below by two other functions. If these bounding functions converge to the same limit at a specific point, the bounded function will also converge to that same limit.

#Formal Statement

Let $f$, $g$, and $h$ be functions defined on an open interval containing $a$ such that:

• $g(x) \le f(x) \le h(x)$ for all $x$ in the interval, and
• $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$

Then: $$\lim_{x \to a} f(x) = L$$

#Worked Example

Let $f(x) = x^2 - 6x + 13$ and $g(x) = 6x - x^2 - 5$. It is known that $g(x) \le f(x)$ for 0 < x < 6. Let $h(x)$ be another function such that 0 < x < 6. Find $\lim_{x \to 3} h(x)$.

Solution: First, find the limits for $f(x)$ and $g(x)$ at $x = 3$ using substitution: $$\lim_{x \to 3} f(x) = (3)^2 - 6(3) + 13 = 4$$