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?
Have you graphed it on your calc?
To determine if a function is continuous on a given interval, we need to check three conditions:
1. The function must be defined on the interval.
2. The limit of the function as it approaches each endpoint of the interval must exist.
3. The value of the function at each endpoint must equal the limit as it approaches that endpoint.
For the function k(t) = (e^t) / (e^t - 7) on the interval [-7,7], let's check each condition.
1. The function is defined for all values of t except when the denominator e^t - 7 is equal to zero. So, let's solve e^t - 7 = 0:
e^t = 7
t = ln(7)
Since ln(7) is a real number, the function is defined on the interval [-7,7].
2. Let's calculate the limit of the function as t approaches the endpoints:
- As t approaches -7: lim(t->-7) (e^t) / (e^t - 7) = (e^-7) / (e^-7 - 7)
- As t approaches 7: lim(t->7) (e^t) / (e^t - 7) = (e^7) / (e^7 - 7)
3. Now, we need to evaluate the function at the endpoints and compare it with the limits:
- k(-7) = (e^-7) / (e^-7 - 7)
- k(7) = (e^7) / (e^7 - 7)
By comparing the limits and the function values at the endpoints, we can conclude if the function is continuous on the interval.
Since all the conditions are met and the function is defined for the interval [-7,7], we can conclude that the function k(t) = (e^t) / (e^t - 7) is continuous on the interval [-7,7]. Therefore, type "continuous" to indicate that the function is continuous on the given interval.