Limite exercise

Limit exercises

See the page on calculation of limits to review how to calculate different types of limits.
Note: the natural logarithm of x is written Lx, instead of ln(x).

1. 
   
   
2. 
   Indetermination 0/0.
   We apply the trick explained here.
   
3. 
   Indetermination 0/0.
   We express each polynomial as a product and simplify the common factors in the numerator and the denominator.
   
   To factorize the polynomials we use Ruffini’s rule:
   
4. 
   Indetermination 0/0.
   The polynomials have a common factor that we can remove:
   
5. Indetermination 0/0.
   We apply equivalences.
   (It can also be solved by applying L'Hôpital).
   
6. Indetermination 0/0.
   It can be solved by applying equivalences:
   
7. Undetermined 0/0.
   It can be solved by applying equivalences:
   
8. Undetermined 0/0.
   It can be solved by applying equivalences:
   
9. Undetermined 0/0.
   It can be solved by applying equivalences.
   
   
10. Undetermined 1^inf
    (See the page on calculation of limits to remember how to solve this type of indetermination).
    
11. Undetermined 0.inf
    We solve it by comparison of infinites:
    
12. Indetermination 0/0.
    We can solve it by applying equivalences, too.
    
13. Undetermined 1^inf
    (See the page on calculation of limits to remember how to solve this type of indetermination).
    
14. Indetermination 0/0
    Let's apply L'Hôpital. (See the page on calculation of limits to remember how to apply it).
    Also, we can apply equivalences:
    
15. Indetermination 0⁰
    (See the page on calculation of limits to remember how to solve this type of indetermination).
    (We applied comparison of infinites).
    

Useful Properties of Logarithms

log(a/b) = log a - log b      

log a.b = log a + log b

log(a - b) = log(a/eb)          

log ak = klog a

               logc a
logb a = ---------
               logc b