Find the limit (if it exists). (If an answer does not exist, enter DNE.)
lim_(t->2) (t^2 + 7 t - 18)/(t^2 - 4)
To find the limit of a function as t approaches a certain value, we can start by directly substituting the given value into the function. So, let's substitute t=2 into the function:
(t^2 + 7t - 18)/(t^2 - 4) = (2^2 + 7(2) - 18)/(2^2 - 4) = (4 + 14 - 18)/(4 - 4) = 0/0
We get an indeterminate form of 0/0, which means we cannot determine the limit by directly substituting the value. In such cases, we need to simplify the expression and try to apply other techniques to find the limit.
To simplify the expression, we can factorize the numerator and denominator:
(t^2 + 7t - 18)/(t^2 - 4) = ((t - 2)(t + 9))/((t - 2)(t + 2))
Now, the (t - 2) terms cancel out:
((t - 2)(t + 9))/((t - 2)(t + 2)) = (t + 9)/(t + 2)
Now, we can substitute t=2 into the simplified expression:
(t + 9)/(t + 2) = (2 + 9)/(2 + 2) = 11/4 = 2.75
Therefore, the limit as t approaches 2 of the given function is 2.75.