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Dra. Diana Denys Jiménez Suro        Dr. Antonio Jesús Sánchez Hernández
Campus Estado de México                             Campus Santa Fe                     
2018
2019
2020
Ene.Ene.Feb.Feb.Mar.Mar.Abr.Abr.May.May.24024020020016016012012080804040
201820192020
Ene.7590120
Feb.150100110
Mar.11515065
Abr.93110140
May.175120210
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Limits and continuity
Mathematical Thinking I
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Dra. Diana Denys Jiménez Suro Dr. Antonio Jesús Sánchez HernándezCampus Estado de México Campus Santa Fe

Limits and continuity

Mathematical Thinking I

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.

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

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

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

¿How do we calculate the limit of the function?

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

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

Continuity at a Point

Jump discontinuity

Removable discontinuity

Removable discontinuity

Discontinuities

Oscillating discontinuity

Infinite discontinuity

Discontinuities

Stewart

Calculus Early Transcendentals8a. Edición, CENGAGE

Thomas

Cálculus Multivariable13a. Edición, PEARSON

Thomas

2018

2015

2015

References

Calculus Early Transcendentals13a. Edición, PEARSON

Antonio JesúsSánchez HernándezCampusSanta Feajsanchez@tec.mx

Diana DenysJiménez SuroCampusEstado de Méxicoddjimenez@tec.mx

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