Limits and continuity
Antonio Jesus Sánche
Created on September 19, 2023
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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
Teachers