evaluate lim x=0 (sin 3x cot 5x)/(xcot 4x)
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.
To evaluate the limit, we can use some trigonometric identities and algebraic simplifications. Let's break down the steps:
Step 1: Rewrite the expression using trigonometric identities.
lim x→0 (sin 3x cot 5x)/(x cot 4x)
= lim x→0 (sin 3x * cos 5x / sin 5x)/(x * cos 4x / sin 4x)
= lim x→0 ((sin 3x * cos 5x) * (sin 4x / (x * cos 4x * sin 5x)))
Step 2: Cancel out the common terms.
lim x→0 (sin 3x * cos 5x) / (cos 4x * sin 5x)
Step 3: Break down all the trigonometric functions.
lim x→0 ((3x) / x) * ((cos 5x) / (cos 4x)) * (sin 4x / (5x) * (sin 5x))
Step 4: Simplify the expression.
lim x→0 3 * (cos 5x) / (cos 4x) * (sin 4x) * (sin 5x) / 5
Step 5: Evaluate the limit by substituting x=0.
lim x→0 3 * (cos 0) / (cos 0) * (sin 0) * (sin 0) / 5
= 3 * 1 * 0 * 0 / 5
= 0
Therefore, the value of the given limit is 0.