Limits

Limit

The output (y-value) that a function is approaching as x approaches a specified value.

One-sided limit notation: xn-

Indicates the left side limit.

One-sided limit notation: xn+

Indicates the right side limit.

Limit Existence Theorem

For a limit to exist, it must have equal one-sided limits.

Continuity

For a function to be continuous, the value must exist (f(c) exists), the limit must exist (lim x→c f(x) exists), and they must be equal (lim x→c f(x) = f(c)).

Horizontal Asymptotes (limits)

Limits as x approaches infinity or negative infinity.

Vertical Asymptotes (limits)

Occurs when substituting results in an undefined value.

Jump Discontinuity

Occurs when one-sided limits are different values.

Removable Discontinuity

Occurs when one-sided limits are equal (limit exists) but not equal to the function value.

Infinite (essential) Discontinuity

Occurs when there is a vertical asymptote.

Top heavy limit

When the degree of the numerator is greater than the degree of the denominator.

Bottom heavy limit

When the degree of the denominator is greater than the degree of the numerator, resulting in a limit of 0.

Equal powers limit approach

If the degrees of the numerator and denominator are equal, the limit is the ratio of the coefficients of the highest powered terms.

Limit definition of the derivative of f(x)

Defined as (f(x+h) - f(x)) / h as h approaches 0.