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

The value that a function approaches as the input approaches a certain value.

An expression whose limit cannot be evaluated directly (e.g., 0/0, ∞/∞).

A method to evaluate limits of indeterminate forms by taking the derivative of the numerator and denominator.

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

A function is continuous at a point if the limit exists at that point, the function is defined at that point, and the limit equals the function value.

A function that can be expressed as the quotient of two polynomials.

A function that is formed by combining two functions, where the output of one function becomes the input of the other.

The process of rewriting an expression using algebraic rules to simplify it or transform it into a more useful form.

An expression formed by changing the sign between two terms in a binomial, often used to rationalize denominators.

The function does not approach a specific value as x approaches a certain point, or the left-hand limit and right-hand limit are not equal.

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.