Find the limit

Question

limit as x--> 0 (8/(xcot5x)

Answer 1

8 / (xcot5x) = 8 tan 5x / x = 8 sin 5x / cos 5x / x.  Rearrange the terms a little to get  (8 / cos 5x) (sin 5x / x).

 

(In everything that follows, "the limit of" means "the limit of, as x --> 0")

 

The limit of (8 / cos 5x) is 8.  (The limit of cos nx is 1, since nx = 0 and cos 0 = 1).

 

The limit of sin nx / x = n.  (That can be proven, but I'll assume that's a fact you already know.  If you want to see it proven, let me know in a comment.)  So the limit of sin 5x / x is 5

 

So the limit of the entire expression is 8 * 5 = 40