COnsider g(x)=(8)/(x-6) on (6,13)
(a) Is this function continuous on the given interval? If it is continuous, type "continuous". If not, give the x -value where the function is not continuous.
Is is continous? If not? what interval then?
no
To determine whether the function g(x) = 8/(x - 6) is continuous on the given interval (6, 13), we need to check two conditions:
1. The function is defined for all x in the interval.
2. The limit of the function as x approaches any value within the interval exists and is equal to the value of the function at that point.
In this case, the function is defined for all x except x = 6 since the denominator (x - 6) would be zero, resulting in an undefined value (division by zero is not allowed). Therefore, we need to investigate the behavior of the function at x = 6 to determine whether it is continuous on the given interval.
To find the limit of the function as x approaches 6, we can evaluate it by substituting 6 into the function:
lim(x->6) g(x) = lim(x->6) 8/(x - 6)
Since the denominator becomes zero when x = 6, we cannot substitute 6 directly into the expression. To work around this, we can factorize the equation using the difference of squares formula:
8/(x - 6) = 8/((√(x - 6)) * (√(x - 6))) = 8/((√(x - 6))^2) = 8/(x - 6)
Now, we can see that the denominator cancels out, resulting in a constant expression:
lim(x->6) 8/(x - 6) = 8/(6 - 6) = 8/0
Here, we encounter an issue because division by zero is undefined. Therefore, the limit does not exist, and the function g(x) is not continuous at x = 6.
In summary, the function g(x) = 8/(x - 6) is not continuous on the interval (6, 13) due to the point of discontinuity at x = 6.