Glossary

Approaching from the Left

Criticality: 2

Examining the behavior of a function as the input values get closer to a specific point from values less than that point. This is denoted by $x→a^-$.

Example:

When analyzing $f(x) = \frac{|x|}{x}$ as x approaches 0, approaching from the left ( $x→0^-$ ) means considering values like -0.1, -0.01, which results in $f(x)$ approaching -1.

Approaching from the Right

Criticality: 2

Examining the behavior of a function as the input values get closer to a specific point from values greater than that point. This is denoted by $x→a^+$.

Example:

For $f(x) = \frac{|x|}{x}$ as x approaches 0, approaching from the right ( $x→0^+$ ) means considering values like 0.1, 0.01, which results in $f(x)$ approaching 1.

Continuity

Criticality: 3

A property of a function where its graph can be drawn without lifting the pen, implying that the limit at a point exists, the function is defined at that point, and the limit equals the function's value.

Example:

The function $f(x) = x^2 + 5$ exhibits continuity for all real numbers, as its graph is a smooth, unbroken parabola.

Direct Substitution

Criticality: 3

A method for evaluating limits where the value 'a' that x approaches is simply plugged into the function, provided the function is continuous at that point and does not result in an indeterminate form.

Example:

To find $\lim\limits_{x→3}(2x+1)$ , we can use direct substitution to get $2(3)+1 = 7$ , since the function is a polynomial and thus continuous.

Discontinuity

Criticality: 2

A point where a function is not continuous, meaning its graph has a break, jump, or hole. Limits are crucial for understanding these points.

Example:

The function $f(x) = \frac{1}{x}$ has a discontinuity at $x=0$ because the function is undefined there and the graph breaks.

Graphical Limit Evaluation

Criticality: 2

Determining a limit by observing the y-value that a function's graph approaches as x gets closer to a specific point.

Example:

By looking at the graph of $f(x) = \frac{1}{x}$ as x approaches infinity, one can use graphical limit evaluation to see the function's y-values approach 0.

Limit

Criticality: 3

The y-value that a function approaches as its input (x-value) gets infinitely closer to a specific point. It describes the function's behavior near a point, not necessarily at the point itself.

Example:

For the function $f(x) = x^2$ , as x gets closer to 2, the function's value gets closer to 4, so the limit as x approaches 2 is 4.

Limit Notation

Criticality: 3

The standard mathematical way to express a limit, written as $\lim\limits_{x→a}f(x) = C$, meaning 'the limit of f(x) as x approaches a equals C'.

Example:

The limit notation $\lim\limits_{x→0}\frac{\sin x}{x} = 1$ concisely states that as x gets very close to 0, the value of $\frac{\sin x}{x}$ approaches 1.

Numerical Limit Evaluation

Criticality: 2

Determining a limit by observing the trend of function values in a table as the input approaches a specific point from both sides.

Example:

To find $\lim\limits_{x→0}\frac{\sin x}{x}$ , one could use numerical limit evaluation by plugging in values like $\pm 0.1, \pm 0.01, \pm 0.001$ and observing that $f(x)$ gets closer to 1.