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

A hole indicates that the function is not defined at that x-value, but the limit may still exist if the function approaches a specific y-value.

A vertical asymptote indicates that the limit approaches infinity (or negative infinity) as x approaches a certain value from one or both sides.

It suggests that the limit does not exist because the function does not approach a single value.

Examine the behavior of the function as x approaches the value from the left and from the right separately.

The graph must not have any breaks, holes, or jumps at that point. You should be able to draw the graph through that point without lifting your pen.

You can see the function being 'squeezed' between two other functions that converge to the same limit.

The graph visually shows that as x approaches 0, the function approaches 1, even though the function is undefined at x = 0.

The slope of the tangent line at a point is the limit of the difference quotient as h approaches 0, which represents the derivative at that point.

Look for vertical asymptotes. If the function approaches infinity (or negative infinity) as x approaches a certain value, then the limit is infinite.

The limit does not exist at a jump discontinuity because the left-hand limit and the 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.