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Glossary

A

Algebraic Manipulation

Criticality: 2

Techniques such as factoring, rationalizing, or finding a common denominator used to rewrite expressions to evaluate limits that initially result in indeterminate forms.

Example:

To evaluate lim⁡x→2x2−4x−2\lim_{x\to 2} \frac{x^2-4}{x-2}, we use algebraic manipulation by factoring the numerator to (x−2)(x+2)(x-2)(x+2) and canceling the common term.

Algebraic Properties of Limits

Criticality: 1

Rules that allow limits of sums, differences, products, quotients, and powers of functions to be evaluated based on the limits of individual functions.

Example:

Using the algebraic properties of limits, we can say that lim⁡x→2(x2+3x)\lim_{x\to 2} (x^2 + 3x) is equal to (lim⁡x→2x2)+(lim⁡x→23x)(\lim_{x\to 2} x^2) + (\lim_{x\to 2} 3x).

D

Discontinuities

Criticality: 2

Points where a function is undefined or behaves erratically, meaning its graph has a break, hole, or jump.

Example:

A piecewise function that suddenly jumps from one y-value to another at a specific x-value exhibits a discontinuity.

I

Infinite Limits

Criticality: 3

Limits that evaluate to positive or negative infinity, indicating that the function's value grows or shrinks without bound as x approaches a certain value.

Example:

When evaluating lim⁡x→0+1x\lim_{x\to 0^+} \frac{1}{x}, the result is ∞\infty, which is an infinite limit.

L

Left-hand Limit

Criticality: 2

The value a function approaches as x gets closer to a specific point from values less than that point.

Example:

For f(x)=∣x∣xf(x) = \frac{|x|}{x}, the left-hand limit as xx approaches 0 is -1, as xx values like -0.001 make the function -1.

Limit Notation

Criticality: 1

The symbolic representation used to express the behavior of a function as its input approaches a certain value.

Example:

The expression lim⁡x→cf(x)=L\lim_{x\to c} f(x) = L is the standard limit notation indicating that as x approaches c, the function f(x) approaches L.

R

Removable Discontinuity

Criticality: 2

A type of discontinuity where a function has a 'hole' in its graph, which can be 'filled' by redefining the function at that single point.

Example:

The function f(x)=x2−1x−1f(x) = \frac{x^2-1}{x-1} has a removable discontinuity at x=1x=1 because the (x−1)(x-1) terms cancel, leaving a hole in the graph.

Right-hand Limit

Criticality: 2

The value a function approaches as x gets closer to a specific point from values greater than that point.

Example:

For f(x)=∣x∣xf(x) = \frac{|x|}{x}, the right-hand limit as xx approaches 0 is 1, as xx values like 0.001 make the function 1.

V

Vertical Asymptotes

Criticality: 3

Vertical lines that a function approaches but never touches, indicating an x-value where the function's behavior is unbounded.

Example:

The function f(x)=1x−5f(x) = \frac{1}{x-5} has a vertical asymptote at x=5x=5, meaning the graph gets infinitely close to this line but never crosses it.