One-sided limit

Definition

A one-sided limit is the value that a function approaches as the input approaches a given value from one side—either from the left or the right. It is denoted as $\lim_{{x \to c^-}} f(x)$ for the left-hand limit and $\lim_{{x \to c^+}} f(x)$ for the right-hand limit.

5 Must Know Facts For Your Next Test

1. The left-hand limit, $\lim_{{x \to c^-}} f(x)$, considers values of $f(x)$ as $x$ approaches $c$ from the left.
2. The right-hand limit, $\lim_{{x \to c^+}} f(x)$, considers values of $f(x)$ as $x$ approaches $c$ from the right.
3. For a general limit to exist at point $c$, both one-sided limits must exist and be equal: $$\lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x)$$.
4. One-sided limits are often used to define continuity at endpoints of intervals.
5. Discontinuities in functions can often be analyzed using one-sided limits to understand behavior near points where standard limits do not exist.

Review Questions

• What is the difference between a left-hand limit and a right-hand limit?
• How do you denote a right-hand limit using mathematical notation?
• Under what condition do both one-sided limits need to be equal for a general limit to exist?

Related terms

Limit:

The value that a function or sequence 'approaches' as the input or index approaches some value.

Continuity:

A function is continuous at a point if its value at that point equals its limit at that point.

Discontinuity:

A point at which a function is not continuous; it can be classified into different types like jump discontinuity or infinite discontinuity.