Question: Lim x -->1 cos2x/(π-2x)^2? Explain in detail.

Solution:

To find the limit of the given expression as x approaches 1, we need to simplify the expression and then substitute the value of x into it. Let's follow the steps below:

Simplifying the expression:
We can simplify the expression by using the trigonometric identity:

cos(2x) = 1 - 2sin^2(x)

Now, let's substitute this identity into our expression:

cos2x/(π-2x)^2 = (1 - 2sin^2(x))/((π-2x)^2)

Applying the limit:
To apply the limit, we substitute the value of x into the expression. In this case, x is approaching 1, so let's substitute x = 1:

[(1 - 2sin^2(1))/((π-2*1)^2)]

Evaluating the expression:
Now, let's evaluate the expression using the given values:

[(1 - 2sin^2(1))/(π-2)^2]

Final Answer: The limit of the expression as x approaches 1 is [(1 - 2sin^2(1))/(π-2)^2].