Evaluate limit as x approaches 0 of:
csc^2(7x)tan(2x^2)
well, we all know that as x->0,
x/sinx -> 1
tanx/x -> 1
csc^2(7x) = (7x)^2 / (7x^2)sin^2(7x)
tan(2x^2) = (2x^2)tan(2x^2) / (2x^2)
multiply those together, and recognizing the limits of 1, you wind up with
(2x^2)/(7x)^2 = 2/49
Thanks
To evaluate the limit as x approaches 0 of the given expression:
lim x→0 csc^2(7x)tan(2x^2)
We'll need to use some trigonometric identities and properties to simplify the expression. Let's break it down step by step:
First, recall the definition of the cosecant (csc) function:
csc(x) = 1/sin(x)
Using this, we can rewrite the expression as:
lim x→0 (1/sin(7x))^2 * tan(2x^2)
Next, let's rewrite the tangent (tan) function using the sine (sin) function:
tan(x) = sin(x) / cos(x)
We can substitute this back into the expression:
lim x→0 (1/sin(7x))^2 * (sin(2x^2) / cos(2x^2))
Now, let's apply the property of limits known as the product rule. The product rule states that the limit of a product is equal to the product of the limits (if they exist). So, we can split the expression into two separate limits:
lim x→0 (1/sin(7x))^2 * lim x→0 (sin(2x^2) / cos(2x^2))
Let's evaluate each limit separately:
1. lim x→0 (1/sin(7x))^2:
As x approaches 0, the value of 7x also approaches 0. We can rewrite the limit as:
lim u→0 (1/sin(u))^2, where u = 7x
Now, this limit is more recognizable. The reciprocal of the sine function is the cosecant function:
lim u→0 csc^2(u)
The limit of csc^2(u) as u approaches 0 is equal to 1. So, we have:
lim u→0 csc^2(u) = 1
Therefore, the first part of the expression simplifies to 1.
2. lim x→0 (sin(2x^2) / cos(2x^2)):
To evaluate this limit, we can recognize that as x approaches 0, both sin(2x^2) and cos(2x^2) approach 0. Applying L'Hopital's Rule, we can take the derivative of the numerator and the denominator with respect to x:
lim x→0 (2cos(2x^2) * 4x) / (-2sin(2x^2) * 4x)
Simplifying further:
lim x→0 (8xcos(2x^2)) / (-8xsin(2x^2))
Now, we can cancel out the common factors of 8x:
lim x→0 (cos(2x^2)) / (-sin(2x^2))
As x approaches 0, the value of 2x^2 also approaches 0. We can rewrite the limit as:
lim u→0 (cos(u)) / (-sin(u)), where u = 2x^2
This is the derivative of the tangent function:
lim u→0 -tan(u)
The limit of -tan(u) as u approaches 0 is equal to 0. So, we have:
lim u→0 -tan(u) = 0
Therefore, the second part of the expression simplifies to 0.
Now, we can combine the simplifications:
lim x→0 (1/sin(7x))^2 * (sin(2x^2) / cos(2x^2)) = 1 * 0 = 0
Hence, the limit as x approaches 0 of the given expression is 0.