Find a value of the constant k such that the limit exists.
x^2-k^2
lim _________
x->8 x-8
k=+/_
It is supposed to be
x^2-k^2 over x-8
And the limit is 8
k=+/- 8
I figured it out.
To find the value of the constant k such that the limit exists, we need to determine if the expression inside the limit approaches a finite value as x approaches 8.
To simplify the expression, let's factor the numerator:
x^2 - k^2 = (x - k)(x + k)
Now let's rewrite the expression with the factored numerator:
lim(x->8) [(x - k)(x + k)] / (x - 8)
To make it easier to evaluate the limit, we can cancel out the factors of (x - 8):
lim(x->8) (x + k)
Now we can directly substitute x = 8 into the expression:
lim(x->8) (8 + k) = 8 + k
For the limit to exist, the expression should approach a finite value that is the same regardless of the approach. In other words, the right-hand limit and the left-hand limit should be equal. Therefore:
lim(x->8+) (8 + k) = lim(x->8-) (8 + k)
Since these limits need to be equal, we can equate them:
8 + k = 8 + k
This equation is true for any value of k. Thus, the value of k can be any real number.