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?
It depends on whether the round brackets surrounding your interval of (6,13) indicate that the numbers 6 and 13 are not included in the interval. The notation I'm familiar with uses square brackets to indicate that the numbers ARE included, and round brackets to indicate that they're NOT. On that basis, the interval (6,13) would NOT include the value 6, in which case the function is continuous over the given range. Had the interval read [6,13], to me this would mean that the interval DOES include 6, and that therefore the function is discontinuous at x=6 (since g(6)=8/0, which is undefined). Whether your brackets mean the same to you as they would to be I have no idea: you would need to check your definitions.
To determine if the function is continuous on the given interval (6, 13), we need to check if it satisfies three conditions:
1. The function must be defined on the entire interval (6, 13). In this case, the function g(x) = 8/(x-6) is defined for all x-values within the interval, excluding x = 6 (since the denominator cannot be zero).
2. The limit of the function as x approaches the endpoints of the interval must exist and be finite. In this case, we need to check if the limit of g(x) as x approaches 6 and 13 exists and is finite.
To determine the limit as x approaches 6, we can substitute the value of x into the function and simplify:
lim(x→6) g(x) = lim(x→6) 8/(x-6) = 8/(6-6) = 8/0 (undefined)
The limit is undefined when x approaches 6 because the denominator becomes zero. Therefore, the function is not continuous at x = 6.
To determine the limit as x approaches 13, we can substitute the value of x into the function and simplify:
lim(x→13) g(x) = lim(x→13) 8/(x-6) = 8/(13-6) = 8/7
The limit exists and is finite when x approaches 13, satisfying the second condition.
Since the function is not continuous at x = 6, the answer to the question "Is this function continuous on the given interval?" is "not continuous at x = 6."