The given function is defined for all
x > 0 except at
x = 8.
Find the value that should be assigned to f(8), if any, to guarantee that f will be continuous at 8. (If an answer does not exist, enter DNE.)
f(x)= Sqrt( x^2-64/x-8)
limit √( (x^2 - 64)/(x-8)) as x -->8
= lim √( (x-8)x+8)/(x-8) , as x ---> 8
= lim √(x+8) , as x---> 8
= √16
= 4
To determine the value that should be assigned to f(8) to guarantee that f will be continuous at x = 8, we need to evaluate the limit of the function as x approaches 8 from both sides and see if the values match.
1. Let's first determine the limit of the function as x approaches 8 from the left side (denoted as x → 8^-):
lim(x → 8^-) sqrt(x^2 - 64 / x - 8)
2. Substitute x = 8 into the function and simplify:
sqrt(8^2 - 64 / 8 - 8)
= sqrt(64 - 64 / 8 - 8)
= sqrt(0 / 0)
3. The result is an indeterminate form (0/0), which means we can't immediately determine the limit. We need to simplify the function further.
4. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
sqrt(0 / 0) * (x + 8) / (x + 8)
= sqrt(0 * (x + 8)^2) / (x + 8)
= sqrt(0) / (x + 8)
= 0 / (x + 8)
= 0
5. We obtained a limit of 0 as x approaches 8 from the left side.
6. Now, let's determine the limit of the function as x approaches 8 from the right side (denoted as x → 8^+):
lim(x → 8^+) sqrt(x^2 - 64 / x - 8)
7. Substitute x = 8 into the function and simplify:
sqrt(8^2 - 64 / 8 - 8)
= sqrt(64 - 64 / 8 - 8)
= sqrt(0 / 0)
8. As in step 3, we have an indeterminate form (0/0).
9. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
sqrt(0 / 0) * (x - 8) / (x - 8)
= sqrt(0 * (x - 8)^2) / (x - 8)
= sqrt(0) / (x - 8)
= 0 / (x - 8)
= 0
10. We obtained a limit of 0 as x approaches 8 from the right side.
11. Since the limits from both sides match, we can assign the value of the limits (which is 0) to f(8) to guarantee the continuity of the function at x = 8.
To summarize, the value assigned to f(8) to guarantee continuity is 0.