Find a value for k so that the function is continuous.
f(x) = {(x^2 + 2x - 15)/(x - 3), x ≠ 3 and 2kx - 4, x = 3
Is k = 2?
yes, k=2
To find a value for k that makes the function continuous, we need to ensure that the limit of the function as x approaches 3 from both sides is the same.
Let's first check the limit as x approaches 3 from the left side:
lim(x->3-) [(x^2 + 2x - 15)/(x - 3)]
To evaluate this limit, we can directly substitute x = 3 into the expression:
lim(x->3-) [(3^2 + 2(3) - 15)/(3 - 3)]
lim(x->3-) [(9 + 6 - 15)/(0)]
lim(x->3-) [0/0]
We get an indeterminate form, which means we need to further simplify the expression to evaluate the limit. To do that, we factorize the numerator:
lim(x->3-) [(x - 3)(x + 5)/(x - 3)]
Now, we can cancel out the common factor of (x - 3):
lim(x->3-) [(x + 5)]
Substituting x = 3 into the expression:
lim(x->3-) [(3 + 5)]
lim(x->3-) [8]
So, the limit of the function as x approaches 3 from the left side is 8.
Now, let's check the limit as x approaches 3 from the right side:
lim(x->3+) [2kx - 4]
Substituting x = 3 into the expression:
lim(x->3+) [2k(3) - 4]
lim(x->3+) (6k - 4)
6k - 4
To make the function continuous, the left-hand limit and right-hand limit should be equal. Therefore, we need to solve the equation:
8 = 6k - 4
Adding 4 to both sides:
12 = 6k
Dividing both sides by 6:
k = 2
Therefore, k = 2 makes the function continuous. Hence, your statement is correct.