Limit Existing: Function approaches a specific value. | Function Continuous: Limit exists, function is defined, and limit equals the function value.

Removable: Limit exists, but function value is different or undefined. | Non-Removable: Limit does not exist (e.g., jump, asymptote).

Graphically: Visual approximation, can be less precise. | Numerically: Approximation using tables, precision depends on step size.

One-sided: Limit from the left or right only. | Two-sided: Limit from both left and right must be equal for the limit to exist.

Continuity: Function has no breaks or jumps. | Differentiability: Function has a derivative at every point (smooth curve).

Average: Slope of secant line over an interval. | Instantaneous: Slope of tangent line at a single point (derivative).

Direct Substitution: Plug in value directly; works if function is continuous. | Algebraic Manipulation: Simplify first (factoring, etc.) when direct substitution fails.

Secant Line: A line that intersects a curve at two points. | Tangent Line: A line that touches a curve at only one point (instantaneous rate of change).

At a Point: Examine function behavior near a specific x-value. | At Infinity: Examine function behavior as x becomes very large or very small.

IVT: Guarantees a specific function value within an interval. | MVT: Guarantees a specific derivative value within an interval.

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

A point on a graph that is not defined, but the limit exists at that point.

The limit exists at the point, the function is defined at the point, and the limit equals the function value at the point.

The value a function approaches as the input approaches a certain value from the left side.

The value a function approaches as the input approaches a certain value from the right side.

For every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.

An expression whose limit cannot be evaluated by direct substitution, such as 0/0 or ∞/∞.

Evaluating a limit by plugging in the value that x approaches into the function.

A function defined by multiple sub-functions, each applying to a certain interval of the domain.

A discontinuity where the limit does not exist, such as a jump or vertical asymptote.

By plugging in values of x that get closer and closer to the target value and observing the trend of f(x).

By visually inspecting the graph and observing where the function is headed as x approaches the target value from both sides.

By using factoring, rationalizing, or trig identities to simplify the function and then using direct substitution.

When the left-hand limit and the right-hand limit are not equal, or when the function approaches infinity.

For a function to be continuous at a point, the limit must exist at that point, the function must be defined at that point, and the limit must equal the function value.

The value a function approaches as the input approaches a certain value from either the left (left-hand limit) or the right (right-hand limit).

If g(x) ≤ f(x) ≤ h(x) for all x near c, and lim x→c g(x) = lim x→c h(x) = L, then lim x→c f(x) = L.

If f is a continuous function on the closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.

Estimating a limit involves approximation using tables or graphs, while finding the exact limit involves algebraic techniques.

To ensure that the limit exists, as both one-sided limits must be equal.