Limits and continuity

Mathematical Thinking I

Limits and continuity

Dra. Diana Denys Jiménez Suro Dr. Antonio Jesús Sánchez HernándezCampus Estado de México Campus Santa Fe

The limit of a function

The concept of a limit is fundamental to finding the tangent to a curve (first intuitively and then formally). We use limits to describe the way a function varies. Some functions vary continuously; small changes in 𝑥 produce only small changes in 𝑓(𝑥). Other functions can have values that jump, vary erratically, or tend to increase or decrease without bound.

What is the behavior of a function when we talk about limits?

Let's consider the function near the point 𝑥=1. Notice that the function is not defined at this point because we would get

¿How do we calculate the limit of the function?

We say that the function 𝑓(𝑥) approaches the limit 𝐿 as 𝑥 tends toby writing In our example, we would say that 𝑓(𝑥) tends to the limit 2 as 𝑥 approaches 1.

Examples

Find the limits

Activity

Canvas

One-Sided Limits

If 𝑓(𝑥) fails to have a limit at 𝑥=𝑎, it may still have a one-sided limit, that is, a limit if the approach is only from one side. If the approach is from the right, the limit is a right-hand limit. From the left, it is a left-hand limit.

Examples of one-side limits

Exercises

Activity

Canvas

Continuity at a Point

A function 𝑓(𝑥) is continuous at a point 𝑥=𝑎 if and only if it meets the following conditions.

Discontinuities

Removable discontinuity

Jump discontinuity

Removable discontinuity

Discontinuities

Infinite discontinuity

Oscillating discontinuity

References

Thomas

Calculus Early Transcendentals13a. Edición, PEARSON

2015

Thomas

Cálculus Multivariable13a. Edición, PEARSON

2015

Stewart

Calculus Early Transcendentals8a. Edición, CENGAGE

2018

Teachers

Antonio Jesús Sánchez Hernández Campus Santa Fe ajsanchez@tec.mx

Diana Denys Jiménez Suro Campus Estado de México ddjimenez@tec.mx