How do you find the points of continuity for F(x) = sin(x^2-2) ?
Can you show your work? Thx
sin(x) is continuous for all real x
x^2-2 is real for all real x
F(x) is continuous everywhere
To find the points of continuity for the function f(x) = sin(x^2-2), we need to check two conditions:
1. The function is defined for all real values of x.
2. The limit of the function exists as x approaches each point in its domain.
Let's first consider the domain of the function. Since the sine function is defined for all real numbers, we only need to determine for which values of x the argument x^2-2 is defined. The argument is defined for all real numbers, so the domain of f(x) is the set of all real numbers.
Now, let's analyze the limit of the function as x approaches each point within its domain. We need to check if the limit exists from both the right and left sides.
For simplicity, let's denote g(x) = x^2 - 2. The function g(x) is defined for all real numbers.
To find the points of continuity, we need to determine if sin(g(x)) has a limit as x approaches each point within its domain. Since sin(g(x)) is a composition of two continuous functions (sin(x) and g(x)), it is continuous whenever g(x) is continuous.
Since g(x) is a polynomial function, it is continuous for all real numbers. Therefore, sin(g(x)) is continuous for all real numbers.
In summary, the function f(x) = sin(x^2-2) is continuous for all real numbers, and there are no points of discontinuity.
Please note that this explanation assumes you have a background in calculus and basic concepts of continuity.