lim x → 0 (x^4) (sec^5(2x)) csc^4(3x)
x csc(3x) = (1/3)((3x)/(sin(3x))
You know that lim u/sinu = 1
So, your limit is
lim sec^5(2x) * (lim (1/3 (3x/sin3x))^4 = 1^5 * (1/3)^4 = 1/81
Limx->0 (5-4/cosx)^1/sin^2(3x)
To find the limit of the given expression, we can use some properties of limits and basic trigonometric identities. Let's break down the expression and simplify it step by step.
First, let's expand the trigonometric functions using their reciprocal identities:
sec(2x) = 1/cos(2x)
csc(3x) = 1/sin(3x)
Our expression becomes:
lim x → 0 (x^4) (1/cos^5(2x)) (1/sin^4(3x))
Next, we can simplify the expression by taking advantage of the fact that cos(0) = 1 and sin(0) = 0:
lim x → 0 (0^4) (1/cos^5(0)) (1/sin^4(0))
Simplifying further, we have:
lim x → 0 (0) (1/1) (1/0^4)
Now we can see that the limit is undefined since we have a division by zero (1/0^4). Therefore, the limit for the given expression does not exist.
In summary, the limit of the expression lim x → 0 (x^4) (sec^5(2x)) csc^4(3x) does not exist because we have a division by zero.