Calculation of a Limit

Limits theory

 

Finite Limit:

 

It is said that the function f(x) has for limit L when “x” moves closer to “a”, and is represented by

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 01

 

Infinite limit:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 02

 
 

PROPERTIES OF LIMITS:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 03

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 04

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 05

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 06

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 07

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 08

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 09

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 10

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 11

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 12

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 13

 
 

THEOREMS:

– The limit exists if and only if the lateral limits exist (for the right and for the left side) and both coincide.

– If there is a limit, it is unique.

 

 

INDETERMINATE FORMS

 

Subtraction

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 14

Solution: We have to multiply and divide by its conjugate.

 
 

Multiplication

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 15

Solution: We do the operation.

 
 

Division

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 16

Solution:

  • In the first case we have three possibilities:
    • Numerator degree > denominator degree. The solution is ± ∞. Example:
Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 17

  • Numerator degree = denominator degree. The solution is the ratio of the coefficients of the same degree. Example:
Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 18

  • Numerator degree < denominator degree. The solution is 0. Example:
BioBioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 19

  • In the second case we have to simplify common factor. In case of being rational roots, we have to multiply and divide by the conjugate of the root.

 
 

Powers

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 20

Solution: We apply the following formula:

BBioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 21

 
 

L’Hôpital’s rule

 

It is a way that uses derivatives to help evaluate limits involving indeterminate forms. This rule is used when we have an uncertainty of type 0/0 or ∞ / ∞.

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Calculation of a Limit

Bioprofe |Calculation of a limit| 22

 

 

 

 

You can download the App BioProfe READER to practice this theory with self-corrected exercises.

 

 

Books that we recommend to extend knowledge of the subject

 

Leave a Reply

Your email address will not be published. Required fields are marked *