# 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 |Calculation of a limit| 01

Infinite limit: Bioprofe |Calculation of a limit| 02

PROPERTIES OF LIMITS: Bioprofe |Calculation of a limit| 03 Bioprofe |Calculation of a limit| 04 Bioprofe |Calculation of a limit| 05 Bioprofe |Calculation of a limit| 06 Bioprofe |Calculation of a limit| 07 Bioprofe |Calculation of a limit| 08 Bioprofe |Calculation of a limit| 09 Bioprofe |Calculation of a limit| 10 Bioprofe |Calculation of a limit| 11 Bioprofe |Calculation of a limit| 12 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 |Calculation of a limit| 14

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

Multiplication Bioprofe |Calculation of a limit| 15

Solution: We do the operation.

Division Bioprofe |Calculation of a limit| 16

Solution:

• In the first case we have three possibilities:
• Numerator degree > denominator degree. The solution is ± ∞. Example: Bioprofe |Calculation of a limit| 17

• Numerator degree = denominator degree. The solution is the ratio of the coefficients of the same degree. Example: Bioprofe |Calculation of a limit| 18

• Numerator degree < denominator degree. The solution is 0. Example: 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 |Calculation of a limit| 20

Solution: We apply the following formula: 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 |Calculation of a limit| 22