Answer:
Explanation:
If we try direct substitution we get:
The radian
is a full rotation from 0 radians, in other words it's equivalent to 0 radians.
Using the unit circle as a reference you'll remember when the angle is 0 radians, the y-value is zero, so:
When the angle is:
it's at a half rotation where the y-value is still zero on the unit circle.
So you get the indeterminate form of:
We can use L'Hôpital's rule which states:
if the original limit is in indeterminate form such as the one above.
Taking the derivative of sin(x) we simply use the property that:
For the sin(2x) we can use the chain rule which essentially states:
It deals with composite equations, and in this case we can represent the composite function as:
Using the chain rule we get:
So now we have the limit:
at pi radians the unit circle is at a half rotation where the x/y coordinates are:
so cos(pi) = -1
at 2 * pi radians were basically back at 0 radians where the x/y coordinates are:
so cos(2 * pi) = 1
Now plugging in these values we get the fraction:
which just simplifies to:
which turns out to be the limit!