At what point is the function y=csc(2x) continuous?
Your question should have been ...
"At what point is the function y=csc(2x) NOT continuous?" to make more sense.
Remember that csc A = 1/sinA
since we cannot divide by zero , the csc(2x) is in trouble whenever sin(2x) = 0
look at y = sin(2x) , when is it zero???
http://www.wolframalpha.com/input/?i=plot+y+%3D+sin(2x)+from+0+to+2%CF%80
Note, x would be in radians in my graph
To determine at what point the function y = csc(2x) is continuous, we need to consider the definition of a continuous function and the properties of the cosecant function.
A function is said to be continuous at a point if the three conditions of continuity are satisfied:
1. The function is defined at that point.
2. The limit of the function exists at that point.
3. The value of the function at that point is equal to the limit.
In the case of y = csc(2x), we need to examine the behavior of the cosecant function.
The cosecant function, csc(x), is defined as the reciprocal of the sine function, sin(x). So, csc(x) = 1/sin(x).
The sine function, sin(x), is not defined at values where its denominator is zero, which occur at integer multiples of pi: x = nπ, where n is an integer. At these points, the function approaches positive or negative infinity.
In our case, we have y = csc(2x), so to solve for the points where the function is discontinuous, we need to identify the values of 2x where the sine function (in the denominator) is equal to zero.
Setting sin(2x) equal to zero, we have:
sin(2x) = 0
Solving for x, we find the values where the sine function is zero:
2x = nπ, where n is an integer.
Dividing both sides by 2, we get:
x = nπ/2, where n is an integer.
These are the points at which the function y = csc(2x) is discontinuous.
To find the points where the function is continuous, we need to exclude these values of x. Therefore, the function y = csc(2x) is continuous at all points except x = nπ/2, where n is an integer.
In summary, the function y = csc(2x) is continuous for all values of x, except at x = nπ/2, where n is an integer.