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.

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.