evaluate lim x=0 (Evaluate sin 3x cot 5x/xcot 4x)
Using L'Hopital's rule:
lim x→0 [sin 3x cot 5x/xcot 4x]
= lim x→0 [(3cos 3x cosec^2 5x - 5cosec^2 5x sin 3x)/(cot 4x - xcosec^2 4x)]
= [3cos 0 cosec^2 0 - 5cosec^2 0 sin 0]/cot 0
= 3/1 = 3
Therefore, lim x→0 (sin 3x cot 5x/xcot 4x) = 3.
To evaluate the limit of the expression sin(3x) cot(5x) / x cot(4x) as x approaches 0, we can simplify it step by step:
Step 1: Apply the trigonometric identity cot(x) = cos(x) / sin(x) to rewrite the expression as sin(3x) / x cot(5x) cos(4x).
Step 2: Now, applying another trigonometric identity, cot(x) = 1/tan(x), we can rewrite it as sin(3x) / x (1/tan(5x)) cos(4x).
Step 3: We know that tan(x) = sin(x) / cos(x). So, substituting this identity, we get sin(3x) / x (1/(sin(5x) / cos(5x))) cos(4x).
Step 4: Simplifying further, we have sin(3x) / x (cos(5x) / sin(5x)) cos(4x).
Step 5: Rearranging the terms, we get sin(3x) cos(4x) / x (cos(5x) sin(5x)).
Step 6: Now, we can see that sin(3x) cos(4x) and cos(5x) sin(5x) have a common factor of sin(5x), which cancels out.
Step 7: Hence, the expression simplifies to sin(3x) cos(4x) / x.
Step 8: Now, we need to evaluate the limit as x approaches 0. Let's break it down into two separate limits, sin(3x) and cos(4x) / x.
Step 9: We know that lim (x→0) sin(x) / x = 1. So, the limit of the second term cos(4x) / x = cos(4*0) / 0 = cos(0) / 0 = 1 / 0, which is undefined.
Step 10: Putting it all together, the limit of the given expression as x approaches 0 is undefined.