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

Look at the y-value the graph approaches as x approaches a certain value. The limit is what y is heading toward, not necessarily the value at that point.

Analyze the trend of y-values as x-values get closer to a specific point. Look for patterns and the value y is approaching.

Use limit laws to break down complex functions into simpler ones. Limits work well with addition, subtraction, multiplication, and division. Consider composite functions.

Manipulate the function to eliminate indeterminate forms (like 0/0). Use conjugates, L'Hôpital's rule, or simplifying rational functions.

When you are given a graph. Scan the graph to see what y-value the function approaches as x approaches a certain value.

When you are given a table of x and y values. Look for patterns and see where the y-values are heading as the x-values approach a specific point.

When you see basic limit theorems (sums, differences, products, quotients) or composite functions. Break down the problem into simpler parts.

When you have complex functions that need simplification. Think conjugates, L'Hôpital's Rule, or simplifying rational functions.

If a function is bounded above and below by two other functions that have the same limit at a certain point, then the function in the middle must also have the same limit at that point.

L'Hopital's Rule can only be applied to indeterminate forms (0/0 or ∞/∞). Applying it to other forms will lead to incorrect results.

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.