f(x)= |x−8| / x+8
Is the function is not continuous at x= -8,8 or just at -8.
f(x)= |x−8| / (x+8)
The function is discontinuous only at x = 8 since it will yield a denominator equal to zero:
f(8) = |8-(-8)| /-8+8
f(8) = 16/0
x = 8 is an asymptotic discontinuity.
Hope this helps~ :3
*oops sorry, I mean x = -8.
So not continuous at x=-8 and function is not defined right.
Yep, not continuous at x = -8.
If you graph the function, it approaches (+) and (-) infinity at x = -8.
To prove, we get the lim as x -> (-8)+ (approaching -8 from the right side) and as x -> (-8)- (approaching -8 from the left side)
lim |x−8| / (x+8) as x -> (-8)+
What we do here is get a value close to -8 but greater than -8, for instance -7.99:
= |-7.99−8| / -7.99+8
= |-15.99| / 0.01
= 1599 (which is a large number and (+))
lim |x−8| / (x+8) as x -> (-8)-
What we do here is get a value close to -8 but less than -8, for instance -8.01:
= |-8.01−8| / -8.01+8
= |-16.01| / -0.01
= -1601 (which is numerically large but (-))
Hope this helps~ :3
To determine whether the function f(x) = |x-8| / (x+8) is continuous at x = -8 and x = 8, we need to check if the function satisfies the conditions for continuity.
A function is said to be continuous at a specific point if three conditions are met:
1. The function is defined at that point (i.e., there is no division by zero).
2. The limit of the function as it approaches the point exists (i.e., the left-hand and right-hand limits are equal).
3. The value of the function at that point is equal to the limit.
Let's analyze these conditions for x = -8:
1. The function is defined at x = -8. In this case, the denominator becomes (x+8) = (-8+8) = 0. Division by zero is not defined, so the function is not defined at x = -8.
Since the function is not defined at x = -8, we can conclude that the function is not continuous at this point.
Now let's consider x = 8:
1. The function is defined at x = 8. In this case, the denominator becomes (x+8) = (8+8) = 16. Division is defined in this case, so the function is defined at x = 8.
To check the continuity at x = 8, we need to evaluate the limit of the function as x approaches 8.
Lim(x->8) [ |x-8| / (x+8) ] = Lim(x->8) [ |8-8| / (8+8) ] = Lim(x->8) [ 0 / 16 ] = 0.
The limit evaluates to 0, and the function is defined at x = 8. Therefore, the limit condition is satisfied.
Since the function is defined at x = 8, the limit exists and the value of the function at x = 8 is also 0, we can conclude that the function is continuous at x = 8.
In summary, the function f(x) = |x-8| / (x+8) is not continuous at x = -8, but it is continuous at x = 8.