1. For what value(s) of x is f(x) = (x-1) / (x^2 - 1) discontinuous?
2.) f(x) = (1)/(x^2) is continuous for all real numbers EXCEPT:
a. x=0
b. x=1 only
c. x=1 and x=-1
d. x=-1 only
e. x=2
look at example 2:
1. hint: removable discontinuity
http://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx
To determine the values of x for which a function is discontinuous, we need to identify any potential points of discontinuity.
1. For the function f(x) = (x-1) / (x^2 - 1), we need to consider the denominator since division by zero causes discontinuity. Therefore, we need to find the values of x that make the denominator zero, as well as any other data points where the function could be discontinuous.
To find the values that make the denominator zero, we set it equal to zero and solve for x:
x^2 - 1 = 0
(x - 1)(x + 1) = 0
From this equation, we can see that the denominator is zero when x = 1 or x = -1.
Now we need to check if these values of x also make the numerator zero. If either value makes the numerator zero, we will have a removable discontinuity, otherwise it will be non-removable.
When x = 1:
f(1) = (1-1)/(1^2-1) = 0/0 (indeterminate form)
When x = -1:
f(-1) = (-1-1)/((-1)^2-1) = -2/0 (undefined)
Therefore, the function f(x) = (x-1) / (x^2 - 1) is discontinuous at x = 1 (removable discontinuity) and x = -1 (non-removable discontinuity).
2. For the function f(x) = 1 / (x^2), we need to determine the values of x for which it is not continuous.
This function involves only division, but the denominator is never zero, so there is no obvious point of discontinuity.
To identify where the function is discontinuous, we need to consider any potential vertical asymptotes or holes in the graph.
The only potential issue is at x = 0 because it could be a vertical asymptote or a hole.
To check if there is a hole, find the limit of the function as x approaches 0:
lim(x→0) 1 / (x^2) = 1 / 0^2 = 1/0 (undefined)
Since the limit is undefined, there is no hole at x = 0.
Therefore, the function f(x) = 1 / (x^2) is continuous for all real numbers except x = 0.
The correct answer is (a) x = 0.