Expert: Paul Klarreich - 1/31/2010

Question
Evaluate the following limit:
Lim x--> pi/x
[sec(x)-tan(x)]

Multiple choice answers are as follows:

A) -1     B) 0       C)  1    D) pi/2         E) Nonexistent


I'm stuck because I don't know how to manipulate the problem in order to use L'Hopitals. I keep getting 1/0 -1/0.

Answer
Questioner: Katherine
Country: United States
Category: Calculus
Private: No
Subject: Limits
Question: Evaluate the following limit:
Lim x--> pi/x

>>>>>>>> Did you mean  pi/2 ?  I really wish questioners would proofread.

[sec(x)-tan(x)]

Multiple choice answers are as follows:

A) -1     B) 0       C)  1    D) pi/2         E) Nonexistent


I'm stuck because I don't know how to manipulate the problem in order to use L'Hopitals. I keep getting 1/0 -1/0.
........................................

Of course you couldn't use it.  It requires that you have the form of

P(x)
----
Q(x)

and you have sec(x) - tan(x).

Since you must have a quotient, you apply not L'Hopital's rule, but Eastwood's rule (that is Clint Eastwood, who said:  "Man's gotta do what a man's gotta do." -- Fistful of Dollars, 1977)

So you just have to make it a quotient.  How about:

sec(x) - tan(x) =
  1      sin x
------ - -------  =
cos x     cos x

1 - sin x
-----------
  cos x

Now you see lim x->pi/2   (that is what you meant, right?)

gives:
1 - 1
----- = 0/0, which is great.  (sort of)
 0

NOW apply l'Hospital's rule.  (and show some respect for the man -- spell it right.)

1 - sin x         - cos x      cos pi/2     0
-----------  --->  ------- -->  --------- = --- = 0
  cos x           - sin x      sin pi/2     1