Understanding calculus can be tricky, especially when it comes to limits and continuity. But there are some important ideas, known as theorems, that can help connect these two topics. Learning these theorems can improve your math skills and make solving problems easier. Let’s break down these important ideas.

1. The Limit Definition of Continuity

A function, or $f(x)$, is continuous at a point called $c$ if it meets three requirements:

1. The function exists at $c$: This means $f(c)$ must be a defined number.
2. The limit exists: When we look at values getting close to $c$, the limit $\lim_{x \to c} f(x)$ needs to be a real number.
3. The limit equals the function value: The limit should match the function value, so it means $\lim_{x \to c} f(x) = f(c)$.

If any of these conditions aren't met, the function is not continuous at that point.

For example, look at the function $f(x) = \frac{x^2 - 1}{x - 1}$. This function doesn’t work at $x = 1$, but as $x$ gets close to 1, it gets closer to 2. So, it’s not continuous at $x = 1$.

2. Squeeze Theorem

The Squeeze Theorem helps us find limits when we can’t just plug in the number. It says that if you have three functions, $f(x)$, $g(x)$, and $h(x)$, and they are set up like this:

$f(x) \leq g(x) \leq h(x)$

for values near $c$, and both $f(x)$ and $h(x)$ have the same limit $L$:

$\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L$

then $g(x)$ must also have the same limit:

$\lim_{x \to c} g(x) = L$

Example: Let’s find the limit of $g(x) = x^2$ as $x$ gets close to 0. We can use $f(x) = 0$ and $h(x) = x^2$. As $x$ approaches 0, both bounds also approach 0, so we can say $\lim_{x \to 0} x^2 = 0$.

3. Limit Laws

Limit Laws are rules that help us solve limits in a structured way. Here are some key rules:

• Sum Law: If $\lim_{x \to c} f(x) = L_1$ and $\lim_{x \to c} g(x) = L_2$, then $\lim_{x \to c} (f(x) + g(x)) = L_1 + L_2$

• Product Law: If the limits exist, then $\lim_{x \to c} (f(x) \cdot g(x)) = L_1 \cdot L_2$

• Quotient Law: If $g(c) \neq 0$, $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L_1}{L_2}$

These rules make it easier to break down tough math problems into smaller parts.

4. Intermediate Value Theorem

The Intermediate Value Theorem (IVT) tells us that if $f(x)$ is continuous on the range from $[a, b]$, and $N$ is any number between $f(a)$ and $f(b)$, then there is at least one point $c$ in the range $(a, b)$ so that $f(c) = N$. This is an important idea for finding solutions in a certain range.

By learning these theorems and ideas, you’ll get a better grip on limits and continuity. This will also set you up for more advanced topics in calculus later on. Remember, practicing with examples and picturing the functions can really help you understand better!