evaluate lim x=0 (sin 3x cot 5x)/(xcot 4x)

Answer 1

We can begin by factoring the expression:

(sin 3x cot 5x)/(xcot 4x) = (sin 3x/sin 4x)(cos 4x/cot 5x)(1/x)

Next, we can use the following limiting values:

lim x→0 sin ax / ax = 1
lim x→0 cos ax - 1 / x = 0
lim x→0 (1 - cos ax) / x^2 = (a^2) / 2

Applying these limits, we get:

(sin 3x/sin 4x)(cos 4x/cot 5x)(1/x)
= (3/4)(1 - 4x^2/2 + O(x^4))(1 - 5x^2/3 + O(x^4))(1/x)
= (3/4)(1 - 2x^2 + O(x^4))(1 - 5x^2/3 + O(x^4))(1/x)
= (3/4)(1 - 2x^2 - 5x^2/3 + O(x^4))(1/x)
= (3/4)(1 - 4x^2/3 + O(x^4))

Taking the limit as x approaches 0, we get:

lim x→0 (sin 3x cot 5x)/(xcot 4x) = (3/4)

Therefore, the limit of the given expression as x approaches 0 is equal to 3/4.