If $f(x) \le g(x) \le h(x)$ for all x near a, and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

If f is continuous on [a, b], then for any value N between f(a) and f(b), there exists a c in (a, b) such that f(c) = N.

If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum value on that interval.

If $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$ (or both approach infinity), and if $f'(x)$ and $g'(x)$ exist, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$.

If you can bound a function between two other functions that have the same limit, then the function in the middle must also have the same limit.

To show that a continuous function takes on a specific value within a given interval.

To guarantee the existence of maximum and minimum values for a continuous function on a closed interval.

When evaluating limits that result in indeterminate forms such as 0/0 or ∞/∞.

The limit of a constant times a function is the constant times the limit of the function: $\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$

The limit of a sum (or difference) is the sum (or difference) of the limits: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$

$\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$

$\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$

$\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$

If $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$

$\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$

$\lim_{x \to c} k = k$

$\lim_{x \to c} x = c$

If $f(x) \leq g(x) \leq h(x)$ for all x near c, and $\lim_{x \to c} f(x) = L = \lim_{x \to c} h(x)$, then $\lim_{x \to c} g(x) = L$.

Direct Substitution: Plug in the value directly. | Algebraic Manipulation: Simplify the expression before plugging in the value (factoring, conjugates, etc.).

Graph: Visual representation of the function's behavior. | Table: Numerical representation of the function's values.

L'Hôpital's Rule: Used for indeterminate forms by taking derivatives. | Squeeze Theorem: Used to find limits by bounding a function between two others.

One-sided limits: Approach from either the left or the right. | Two-sided limits: Approach from both sides, and both must agree for the limit to exist.

Limit exists: The function approaches a value. | Continuous: The limit exists, the function is defined, and they are equal.

Removable: Can be 'fixed' by redefining the function at a single point. | Non-removable: Cannot be fixed (e.g., jump, asymptote).

Finite values: Focus on the function's behavior near a specific point. | Infinity: Focus on the function's end behavior as x grows without bound.

Limit Laws: Apply basic properties of limits (sum, product, etc.). | Algebraic Manipulation: Transform the expression to simplify it before applying limit laws.

0/0: Both numerator and denominator approach zero. | ∞/∞: Both numerator and denominator approach infinity.

Conjugates: Used for expressions with radicals. | Factoring: Used for polynomial expressions.