lim xcsc(3x)
x-->0
hint
sin x ---> x as x --->0
thanks!
To find the limit of csc(3x) as x approaches 0, we can first simplify the expression.
Recall that csc(x) is equal to 1/sin(x).
So, csc(3x) can be written as 1/sin(3x).
Now, let's evaluate the limit:
lim x→0 of csc(3x) = lim x→0 of 1/sin(3x)
Since sin(0) = 0, we have 1/0 in the denominator, which is undefined.
To further evaluate this limit, we can use a trigonometric identity known as the "sine of double angle":
sin(2θ) = 2sin(θ)cos(θ).
Rearranging this identity, we can write sin(θ) = (1/2)sin(2θ)/cos(θ).
Now, let's apply this trigonometric identity to our expression:
sin(3x) = (1/2)sin(6x)/cos(3x)
So, our limit becomes:
lim x→0 of 1/sin(3x) = lim x→0 of 1/[(1/2)sin(6x)/cos(3x)]
Next, we can simplify the expression by multiplying the numerator and denominator by cos(3x):
lim x→0 of 1/sin(3x) = lim x→0 of cos(3x)/[(1/2)sin(6x)]
Now, we can break down the limit into two separate limits:
lim x→0 of cos(3x) and lim x→0 of [1/2sin(6x)].
Let's evaluate the first limit:
lim x→0 of cos(3x) = cos(0) = 1.
For the second limit, we can use the trigonometric identity sin(2θ) = 2sin(θ)cos(θ) again:
sin(6x) = sin(2(3x)) = 2sin(3x)cos(3x).
Therefore, our second limit becomes:
lim x→0 of [1/2sin(6x)] = lim x→0 of [1/(2sin(3x)cos(3x))]
= lim x→0 of [1/(2 * 2sin(3x) * cos(3x))]
= lim x→0 of [1/(4sin(3x)cos(3x))].
Now, let's evaluate this limit:
lim x→0 of [1/(4sin(3x)cos(3x))] = 1/(4 * 0 * 1) = 0.
So, the overall limit becomes:
lim x→0 of csc(3x) = lim x→0 of 1/sin(3x) = 1/0 = undefined.