Limit

Definition

A limit describes the value that a function approaches as the input approaches some value. It is fundamental in understanding calculus concepts such as continuity, derivatives, and integrals.

5 Must Know Facts For Your Next Test

1. The notation $\lim_{{x \to c}} f(x) = L$ indicates that as $x$ approaches $c$, $f(x)$ approaches $L$.
2. Limits can be evaluated from two sides: the left-hand limit ($\lim_{{x \to c^-}} f(x)$) and the right-hand limit ($\lim_{{x \to c^+}} f(x)$).
3. A function has a limit at a point if and only if both one-sided limits exist and are equal.
4. If a function is continuous at a point, then the limit of the function as it approaches that point is equal to the function's value at that point.
5. Certain techniques for evaluating limits include direct substitution, factoring, rationalizing, and using special limits such as $\lim_{{x \to 0}} \frac{\sin x}{x} = 1$.

Review Questions

• What does it mean for $\lim_{{x \to c}} f(x) = L$?
• How can you determine if a limit exists at a particular point?
• What methods can be used to evaluate limits?

Related terms

Continuity:

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.

Derivative: The derivative of a function measures how the output of the function changes as its input changes; it is defined using limits.

Infinity: $\infty$ represents an unbounded quantity; limits involving infinity describe behavior as values grow larger without bound or decrease without bound.