Determine whether f is continuous at x=c in the function. Use the 3 conditions of continuity to justify your answer.
f(x) = x-1/x+1 c=-1
My answer is not continuous. Am I correct?
correct
To determine whether the function f(x) = (x-1)/(x+1) is continuous at x = -1, we need to check the three conditions of continuity.
1. The function should be defined at x = c.
For the given function, the function is defined for all values of x except x = -1. Therefore, the first condition is not satisfied.
2. The limit of the function as x approaches c should exist.
Let's compute the limit of f(x) as x approaches -1:
lim (x → -1) (x-1)/(x+1)
Using direct substitution, we get:
(-1-1)/(-1+1)
-2/0
Since the denominator is zero, the limit does not exist. Therefore, the second condition is not satisfied.
3. The limit of the function as x approaches c should be equal to the value of the function at x = c.
Since the second condition is not satisfied, we don't need to proceed with this step.
Based on the above analysis, we can conclude that the function f(x) = (x-1)/(x+1) is not continuous at x = -1. Therefore, your answer is correct.
To determine whether the function f(x) = (x-1)/(x+1) is continuous at x = c = -1, we need to evaluate the three conditions of continuity.
The three conditions of continuity are:
1. The function f(c) is defined.
2. The limit of f(x) as x approaches c exists.
3. The limit of f(x) as x approaches c is equal to f(c).
Let's evaluate each condition:
1. The function f(c) is defined:
To evaluate f(c), substitute c = -1 into the function:
f(c) = f(-1) = (-1 - 1)/(-1 + 1) = -2/0
Since the denominator is 0, the function is undefined at x = c = -1. Thus, the first condition for continuity is not satisfied.
2. The limit of f(x) as x approaches c exists:
To evaluate the limit, we need to check if the left-hand limit, LHL, and the right-hand limit, RHL, exist and are equal as x approaches c = -1.
LHL: lim(x->-1-) [(x-1)/(x+1)] = lim(x->-1-) (x-1)/(x+1) = (-1-1)/(-1+1) = -2/0, which is undefined.
RHL: lim(x->-1+) [(x-1)/(x+1)] = lim(x->-1+) (x-1)/(x+1) = (-1-1)/(-1+1) = -2/0, which is undefined.
Since both the left-hand limit and the right-hand limit are undefined as x approaches -1, the second condition is not satisfied.
3. The limit of f(x) as x approaches c is equal to f(c):
Since we couldn't evaluate the limits in the previous step due to undefined expressions, this condition is not applicable.
Based on the above analysis, we can conclude that the function f(x) = (x-1)/(x+1) is not continuous at x = c = -1.
Therefore, your answer that it is not continuous is correct.