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:

1. Direct Substitution: If the function is continuous at the point of interest, simply substitute the point into the function.
2. Factoring: If the function is a rational function with a removable discontinuity, factor the numerator and denominator and cancel common factors.
3. 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.
4. 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:

1. The function is defined at the point.
2. The limit of the function exists at the point.
3. 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).

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.