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

 

Infinite limit:

 
 

PROPERTIES OF LIMITS:

 
 

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

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

 
 

Multiplication

Solution: We do the operation.

 
 

Division

Solution:

• In the first case we have three possibilities:
  • Numerator degree > denominator degree. The solution is ± ∞. Example:

• Numerator degree = denominator degree. The solution is the ratio of the coefficients of the same degree. Example:

• Numerator degree < denominator degree. The solution is 0. Example:

• 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

Solution: We apply the following formula:

 
 

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 ∞ / ∞.

 

 

 

 

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