# Limits and Continuity

## Introduction to Limits and Continuity

Limits and continuity are fundamental concepts in Calculus that describe the behavior of functions as the input values approach a certain point or infinity. Understanding these concepts is crucial for learning more advanced topics, such as derivatives and integrals.

## Common Terms and Definitions

**Limit**: The value that a function approaches as the input values get closer to a specific point.

**One-sided Limit**: The limit of a function as the input values approach a point from either the left or the right side.

**Limit at Infinity**: The value that a function approaches as the input values become arbitrarily large or small.

**Continuity**: A function is continuous at a point if the limit of the function at that point exists and equals the function value at that point.

**Removable Discontinuity**: A discontinuity that can be eliminated by redefining the function value at a single point.

**Jump Discontinuity**: A discontinuity where the function has different left and right limits at a point.

**Infinite Discontinuity**: A discontinuity where the function approaches positive or negative infinity as the input values approach a point from either side.

## Evaluating Limits

To evaluate limits, you can use various techniques, such as:

- Direct Substitution: If the function is continuous at the point of interest, simply substitute the point into the function.
- Factoring: If the function is a rational function with a removable discontinuity, factor the numerator and denominator and cancel common factors.
- Rationalization: If the function contains a square root or other root in the denominator, multiply the numerator and denominator by the conjugate of the denominator.
- L'Hôpital's Rule: If the limit results in an indeterminate form (0/0, ∞/∞, 0 · ∞, ∞ - ∞, 0^0, 1^∞, or ∞^0), differentiate the numerator and denominator separately and evaluate the limit of the resulting quotient.

## Continuity

A function is continuous at a point if it satisfies the following conditions:

- The function is defined at the point.
- The limit of the function exists at the point.
- The limit of the function equals the function value at the point.

To determine the continuity of a function, evaluate the limits from both sides of the point and compare them to the function value at that point.

## Common Questions and Answers

**What is the difference between a limit and a function value?**

A limit describes the behavior of a function as the input values approach a specific point, while a function value is the output of the function at a specific input value. A function can have a limit at a point even if it is not defined at that point.

**How do you evaluate a limit when direct substitution results in an undefined expression?**

When direct substitution results in an undefined expression, such as 0/0 or ∞/∞, you can use techniques like factoring, rationalization, or L'Hôpital's Rule to evaluate the limit.

**What are the different types of discontinuities?**

The three main types of discontinuities are removable discontinuities (can be eliminated by redefining the function at a single point), jump discontinuities (the function has different left and right limits at a point), and infinite discontinuities (the function approaches positive or negative infinity as the input values approach a point from either side).

Get your questions answered instantly by an AI Calculus tutor.## Conclusion

Limits and continuity are essential concepts in Calculus that provide a foundation for understanding more advanced topics. By mastering the techniques for evaluating limits and determining the continuity of functions, you will be well-prepared to tackle a wide range of problems in Calculus and its applications.