CALC KEYWORDS

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Target value or Value of approach

The input value that the function is getting close to.

To approach a value from the left

To move toward a target value using input values that are smaller than the target.

To approach a value from the right

To move toward a target value using input values that are larger than the target.

To approach a value from both sides

To move toward a target value using input values from both smaller and larger sides.

To approach positive infinity

To increase without bound in the positive direction.

To approach negative infinity

To decrease without bound in the negative direction.

Limit

The value that the outputs of a function get close to as the inputs approach the target value.

Limit is finite

The outputs of the function get close to a single real number as the inputs approach the target value.

Limit is positive infinity

The outputs of the function increase without bound as the inputs approach the target value.

Limit is negative infinity

The outputs of the function decrease without bound as the inputs approach the target value.

Limit doesn't exist

The outputs of the function do not settle toward a single value as the inputs approach the target value.

Vertical Asymptote

A vertical line that a graph gets closer to but never reaches, where outputs grow without bound.

Graphical approach

A method of finding limits by visually examining the graph of a function and identifying patterns in how the graph behaves near a target value.

Numerical approach

A method of finding limits by computing outputs for inputs that move closer and closer to the target value and analyzing the resulting patterns.

Input

A chosen value used to evaluate the function.

Output

The corresponding function value produced by an input.

Table of inputs/outputs

A list of input–output pairs used to observe how a function behaves near a target value.

Sequence

An ordered list of numbers.

Sequence of inputs

A sequence of x‑values chosen to approach a target value.

Sequence of outputs

The corresponding list of function values generated by a sequence of inputs.

Zooming in numerically

Focusing on outputs for inputs very close to the target value to reveal the local behavior of the function.

Analytic approach

The method of finding a limit using algebraic manipulation and established mathematical rules (laws) rather than relying on a visual graph or a table of values.

Limit laws

A set of rules that allow you to evaluate the limit of a complex function by breaking it into simpler components.

Direct substitution property

A property of 'well-behaved' functions where the limit as x \rightarrow c is simply the value of the function at x = c, or f(c).

Power functions

Functions of the form f(x) = ax^n where n is a natural number.

Polynomial functions

Functions that can be expressed as sums of power functions.

Reciprocal functions

Functions of the form f(x) = a x^{-n} where n is a natural number.

Rational functions

Functions that can be written as a ratio of two polynomials.

Rational-like functions

Functions that may involve radicals or rational powers but behave similarly to rational functions in their structure.

Exponential functions

A family of functions typically of the form a^x with a > 0.

Logarithmic functions

Inverse functions of exponentials, often written as \log_a(x).

Trigonometric functions

The standard circular functions such as \sin(x), \cos(x), and others.

Radical functions

Functions involving roots, such as \sqrt{x} or x^{(1/n)}.

Piecewise-defined functions

Functions defined by different expressions over different intervals.

Substitution technique

A technique where a complicated expression is replaced with a simpler variable.

Infinite arithmetic

Rules describing how expressions behave when involving positive or negative infinity.

Infinitesimal arithmetic

Rules describing how expressions behave when involving values that approach zero.

Determinate form

An expression where the limit can be found directly using basic arithmetic and limit laws.

Indeterminate form

An expression that does not have a set value and requires further algebraic techniques to evaluate.

The Squeeze Theorem

A geometric/analytic method used to find the limit of a function by 'trapping' it between two other functions whose limits are known and equal.

Special limit

A specific, proven limit result that must be memorized.

Graphical approach

Using the graph of a function to visually determine the value the function approaches.

Numerical approach:

Using tables, sequences, or zooming in numerically to observe how function values behave near a target.

Analytical approach

Using algebraic manipulation, limit laws, and function‑type strategies to compute limits symbolically.

Infinite & infinitesimal arithmetic

Using rules that describe how expressions behave when quantities become very large (approach ±∞) or very small (approach 0).

Formal approach

Using the epsilon–delta definition to quantify how x approaches a target value by expressing closeness with precise inequalities.

Continuous at a number

A function has no break at a value a; the limit as x approaches a exists and equals the function’s value there.

Continuous from the right at a number

A function has no break when approached from values greater than a; the right-hand limit equals the function’s value at a.

Continuous from the left at a number

A function has no break when approached from values less than a; the left-hand limit equals the function’s value at a.

Continuous on an interval

A function behaves without interruption on an interval; one-sided limits apply at endpoints when the interval is closed.

Continuous

A function behaves without interruption everywhere it is defined, i.e., no breaks at any point in its domain.

Continuity

The idea that a function’s graph can be drawn without lifting a pencil, meaning the limit at each point matches the function’s value there.

Removable discontinuity

A situation where the limit at a point exists but does not match the function’s value; often fixable by redefining the function at that point.

Jump discontinuity

A situation where the left-hand and right-hand limits both exist but are unequal, creating a vertical jump.

Infinite discontinuity

A situation where the function grows without bound near a point; at least one one-sided limit is +∞ or −∞, often corresponding to a vertical asymptote.

Intermediate Value Theorem

If a function is continuous on a closed interval [a, b], then it attains every value between f(a) and f(b) at some point in the interval.

Tangent Line

A line that “just touches” a curve at a specific point, representing the instantaneous direction of the curve at that point.

Secant Line

A line that cuts through a curve at two distinct points, (a, f(a)) and (b, f(b)).

Osculating Circle

The “best-fitting” circle at a specific point on a smooth curve.

Curvature

A measure of how tightly a curve bends, defined as the reciprocal of the radius of the osculating circle (κ = 1/R).

Smooth Curve

A curve that has a unique tangent line at every point (no sharp corners or breaks).

Average Rate of Change

The slope of the secant line over an interval [a, b]; it describes how the output changed on average across that span.

Instantaneous Rate of Change

The exact rate at which a function is changing at a single moment; geometrically, this is the slope of the tangent line.