Give a formula for the extended function that is continuous at the indicated point.
F(x)= (x-4)/((√x)-2) at x=4
I know the answer I just need to figure out how to work it out. The answer btw is
y= (√x) + 2
factor the numerator as a difference of two squares.
(x-4)=(sqrtx +2)(sqrtx -2)
now divide out the sqrtx -2
To find the extended function that is continuous at the point x = 4, we can start by determining the limit of the function as x approaches 4.
We can rewrite the function F(x) as:
F(x) = (x - 4) / (√x - 2)
To determine the limit as x approaches 4, substitute x = 4 into the function:
F(4) = (4 - 4) / (√4 - 2)
Simplifying the expression, we have:
F(4) = 0 / (2 - 2)
F(4) = 0 / 0
Since we end up with an indeterminate form of 0/0, we can use L'Hôpital's rule to evaluate the limit.
Differentiating the numerator and the denominator with respect to x, we get:
F'(x) = 1 / (2 * √x)
Applying the derivative to the function, we have:
F'(4) = 1 / (2 * √4)
F'(4) = 1 / (2 * 2)
F'(4) = 1 / 4
Therefore, the limit as x approaches 4 is 1/4.
To make the function continuous at x = 4, we need the limit and the function value at x = 4 to be equal. So we set up the equation:
lim(x→4) F(x) = F(4)
1/4 = (4 - 4) / (√4 - 2)
1/4 = 0 / (2 - 2)
1/4 = 0 / 0
To solve the equation, we can multiply both sides by the denominator (√4 - 2):
(√4 - 2) / (2 - 2) = 0
(√4 - 2) = 0
√4 = 2
Simplifying, we have:
2 = 2
Therefore, the equation that allows F(x) to be continuous at x = 4 is:
F(x) = (√x) + 2
To find the formula for the extended function that is continuous at a specific point, such as x = 4 in this case, you need to evaluate the limits of the function as it approaches the given point from both sides.
Let's start by finding the limit as x approaches 4 from the left side, denoted as "x → 4^−".
To do that, substitute x = 4 - h into the given function, where h is a positive number approaching zero:
F(x) = ((4 - h) - 4)/(√(4 - h) - 2)
Simplifying the numerator:
= -h/(√(4 - h) - 2)
Now simplify the denominator by rationalizing it:
= -h/(√(4 - h) - 2) × (√(4 - h) + 2)/(√(4 - h) + 2)
= -h(√(4 - h) + 2)/(4 - (2^2))
Simplifying further:
= -h(√(4 - h) + 2)/(4 - 4)
= -h(√(4 - h) + 2)/0
As h approaches zero, this expression approaches negative infinity.
Now let's find the limit as x approaches 4 from the right side, denoted as "x → 4^+".
Substitute x = 4 + h into the given function, where h is a positive number approaching zero:
F(x) = ((4 + h) - 4)/(√(4 + h) - 2)
Simplifying the numerator:
= h/(√(4 + h) - 2)
For the denominator, we will rationalize it as before:
= h/(√(4 + h) - 2) × (√(4 + h) + 2)/(√(4 + h) + 2)
= h(√(4 + h) + 2)/(4 + (2^2))
Simplifying further:
= h(√(4 + h) + 2)/(h + 4)
The h terms cancel out:
= √(4 + h) + 2
As h approaches zero, this expression approaches √4 + 2 = 2 + 2 = 4.
Since the limit of F(x) as x approaches 4 from the left side is negative infinity, and the limit from the right side is 4, in order to make the function continuous at x = 4, the value of F(x) at x = 4 must be equal to the right-side limit, which is 4.
Therefore, the formula for the extended function that is continuous at x = 4 is:
F(x) = √x + 2