Expert: Paul Klarreich - 1/31/2011

Question
I'm having some trouble with limit problems. I am hoping that you could give me step by step instructions ( i already know the answers.)

1. Find lim (x->infinity): ((x^4) + 5)/ ((x^3) + (6x^2))
2. Find lim (x->0): ((6-x)^(1/2)) - (6^(1/2)) / x
3. Find lim (x->0): (1-cosx)/ (sinx)

Thank you for your time.

Answer
Questioner: Elinor  
Country: United States
Category: Calculus
Private: No
Subject: Calculus: Limits
Question: I'm having some trouble with limit problems. I am hoping that you could give me step by step instructions ( i already know the answers.)

1. Find lim (x->infinity): ((x^4) + 5)/ ((x^3) + (6x^2))
2. Find lim (x->0): ((6-x)^(1/2)) - (6^(1/2)) / x
3. Find lim (x->0): (1-cosx)/ (sinx)

Thank you for your time.
................................................
You could probably use l'Hospital's rule on these, but:


                             x^4 + 5
1. Find lim (x->infinity): -----------
                           x^3 + 6x^2

Divide each term by your highest power: x^4,  then it is easy.


                    (6-x)^(1/2) - 6^(1/2)
2. Find lim (x->0): ----------------------
                              x

Rationalize:

(6-x)^(1/2) - 6^(1/2)  (6-x)^(1/2) + 6^(1/2)
--------------------- ------------------------
         x             (6-x)^(1/2) + 6^(1/2)


    6 - x - 6
--------------------------
 x [(6-x)^(1/2) + 6^(1/2)]

etc,

                   (1 - cosx)(1 + cosx)
3. Find lim (x->0): -------------------- << Rationalize:
                      sinx (1 + cosx)

Simplify, use an identity, etc.