Calculating Limits


LIMIT LAWS: Suppose that is a constant and the limits



exist. Then

1.

2.

3.

4.

5.    if  

6. where is a positive integer

7.

8.

9. where is a positive integer

10. where is a positive integer (if is even, we assume that )

11. where is a positive integer (if is even, we assume that )

EXAMPLES:








DIRECT SUBSTITUTION PROPERTY: If f is a polynomial or a rational function and a is in the domain of f, then

 \lim_{x \to a} f(x) =f(a)

REMARK: The trigonometric functions also enjoy the Direct Substitution Property.

EXAMPLES:






3.


4.


5.


6.


7.


8.


9.


10.


11.


12.


13.


14.


15.


16.


17.


18.


19.


20.


21.


22. Let



  Find, if possible, and


23. Let



Find, if possible,




24. Show that does not exist.



THE SQUEEZE THEOREM: If when is near (except possibly at ) and



then .

25. Show that