Glossary
Approaching from the Left (a-)
Describes the behavior of a function as the input 'x' gets closer to a specific value 'a' by taking values strictly less than 'a'.
Example:
When a car approaches a stop sign, approaching from the left means observing its speed as it decreases from values just below the stop sign's position.
Approaching from the Right (a+)
Describes the behavior of a function as the input 'x' gets closer to a specific value 'a' by taking values strictly greater than 'a'.
Example:
If you're analyzing the temperature of a cooling object, approaching from the right means looking at temperatures just above the target temperature as time progresses.
Direct Substitution
A method of evaluating a limit by plugging the target value directly into the function's expression.
Example:
To find the limit of f(x) = x + 3 as x approaches 2, you can use direct substitution to get 2 + 3 = 5.
Estimating Limits from Tables
A numerical method to approximate the value of a limit by observing the trend of function outputs (y-values) as inputs (x-values) get progressively closer to the target value from both sides.
Example:
To find the limit of a complex function at x=0, you might use estimating limits from tables by plugging in x-values like -0.01, -0.001, 0.001, and 0.01 to see what y-value the function approaches.
Indeterminate Form
An expression (like 0/0 or ∞/∞) that results from direct substitution into a limit, indicating that the limit cannot be determined by substitution alone and requires further analysis.
Example:
When trying to find the limit of (sin x)/x as x approaches 0, direct substitution yields 0/0, which is an indeterminate form, signaling the need for L'Hôpital's Rule or algebraic manipulation.
Limit
The value that a function approaches as its input (x) gets arbitrarily close to a specific value, regardless of the function's value at that exact point.
Example:
When considering the function f(x) = (x^2 - 4)/(x - 2), as x gets closer and closer to 2, the function's output approaches 4, so the limit as x approaches 2 is 4.
Limit Does Not Exist
A condition where a limit cannot be assigned a single finite value, often because the one-sided limits are not equal, the function oscillates, or it approaches infinity.
Example:
For the function f(x) = |x|/x, the limit does not exist at x=0 because the left-hand limit is -1 and the right-hand limit is 1, which are not equal.
One-Sided Limits
The behavior of a function as its input approaches a specific value from either the left (values less than the target) or the right (values greater than the target).
Example:
For a piecewise function, the one-sided limit from the left at x=0 might be 5, while the one-sided limit from the right at x=0 might be 2, indicating a jump discontinuity.
Vertical Asymptote
A vertical line (x=a) that a function's graph approaches but never touches, typically occurring where the function's value approaches positive or negative infinity.
Example:
The function f(x) = 1/x has a vertical asymptote at x=0 because as x approaches 0, the function's values grow infinitely large or small.