consider k(t)=(e^t)/(e^t-7) on[-7,7]

Is this function continuous on the given interval? If it is continuous, type "continuous". If not, give the t -value where the function is not continuous.

I say continous, am I wrong? If I am wrong what interval is it on?

To determine if the function k(t) = (e^t)/(e^t-7) is continuous on the given interval [-7,7], we need to check three conditions:

1. The function is defined at every point on the interval. In this case, e^t is defined for all real numbers, and e^t ≠ 7 for all t. Therefore, the function is defined at every point on the interval [-7,7].

2. The function has a limit from both the left and the right at every point on the interval. To check this, we need to find the limits of k(t) as t approaches -7 and 7 from both sides.

As t approaches -7 from the left, k(t) = (e^t)/(e^t-7) approaches (e^-7)/(e^-7-7) = 1/(1-7) = -1/6.

As t approaches -7 from the right, k(t) = (e^t)/(e^t-7) approaches (e^-7)/(e^-7-7) = 1/(1-7) = -1/6.

Therefore, the limit of k(t) as t approaches -7 from both sides is the same, which indicates continuity at t = -7.

As t approaches 7 from the left, k(t) = (e^t)/(e^t-7) approaches (e^7)/(e^7-7).

As t approaches 7 from the right, k(t) = (e^t)/(e^t-7) approaches (e^7)/(e^7-7).

Therefore, the limit of k(t) as t approaches 7 from both sides is the same, indicating continuity at t = 7.

3. The function's value matches the limit at every point on the interval. Since the limits as t approaches -7 and 7 from both sides exist and are equal to -1/6 and (e^7)/(e^7-7), respectively, and the function is defined at -7 and 7, the function k(t) = (e^t)/(e^t-7) is continuous on the interval [-7,7].

Thus, your statement "continuous" is correct. The function is continuous on the interval [-7,7].