a) Give an example of a function, whose limit exists as x approaches 2, but it is not continuous at x=2
b)Give an example of a function that is not differentiable at x=2, but it is continuous at x=2
a) One example of a function that has a limit as x approaches 2 but is not continuous at x=2 is the function f(x) = 1/x. Let's examine this function to see its behavior.
To find the limit of f(x) as x approaches 2, we can substitute 2 into the function:
lim(x→2) 1/x = 1/2 = 0.5
As we can see, the limit of f(x) as x approaches 2 exists and equals 0.5. However, the function is not continuous at x=2 because f(2) is undefined. Dividing by zero is not defined in mathematics, so the function f(x) = 1/x is not defined at x=2. Therefore, it is not continuous at x=2 despite having a limit.
b) An example of a function that is continuous at x=2 but not differentiable at x=2 is the function f(x) = |x-2|.
To determine continuity, we need to check if the left-hand limit (LHL) and the right-hand limit (RHL) at x=2 exist and are equal. In this case, both the LHL and RHL exist and are equal to zero:
lim(x→2-) |x-2| = 0
lim(x→2+) |x-2| = 0
Thus, the function is continuous at x=2.
To determine differentiability, we need to check if the derivative of the function exists at x=2. However, the function f(x) = |x-2| has a corner or a sharp bend at x=2, meaning the slope of the function changes abruptly at that point. As a result, the derivative does not exist at x=2, and the function is not differentiable at x=2, despite being continuous.
a) One example of a function that has a limit as x approaches 2, but is not continuous at x = 2 is:
f(x) = {
x, x ≠ 2
0, x = 2
}
In this function, as x approaches 2 from both sides (i.e., x < 2 and x > 2), the limit of f(x) is equal to 2. However, at x = 2, the function has a "jump" in its graph. Therefore, it is not continuous at x = 2 despite the limit existing.
b) One example of a function that is continuous at x = 2 but not differentiable at x = 2 is:
f(x) = |x - 2|
This function represents the absolute value of x - 2. It is continuous at x = 2 because the limit of f(x) as x approaches 2 from both sides (i.e., x < 2 and x > 2) is equal to 0. However, at x = 2, the function has a "corner" in its graph, making it not differentiable at x = 2.