Construct a function so that lim x → 4 f(x) = 2 with the following restrictions: The function is rational, the limit must contain an indeterminate form, the function must contain a radical in the numerator, and it must contain a trinomial in the denominator. Determine such a function
you want the top and bottom to be zero, so how about
2√(3x-8)/((x-2)(x-3))
most rational functions contain only polynomials, so maybe you want to modify this one so that the radical is just a constant.
Boy - I messed that up. No indeterminate form. You need x-4 in both top and bottom
(√(3x-8)(x^2-16))/(8(x-2)(x-3))
AAAaarrrrgghhh! ... I mean
(√(3x-8)(x^2-16))/(8(x-4)(x-3))
To construct a function that satisfies the given restrictions, we can start by creating a fraction with a radical in the numerator and a trinomial in the denominator.
Let's use the following function as an example:
f(x) = (sqrt(x - 4)) / (x^2 - 16)
Here's how we can explain it step by step:
1. Start with a square root in the numerator: sqrt(x - 4).
This satisfies the requirement of having a radical in the numerator.
2. In the denominator, use a trinomial: x^2 - 16.
This is a quadratic polynomial of the form (ax^2 + bx + c) = (x^2 - 16).
3. Substitute x = 4 into the function and check the result:
f(4) = (sqrt(4 - 4)) / (4^2 - 16) = sqrt(0) / 0.
At this point, we have an indeterminate form of 0/0, which we need to resolve in order to find the limit.
4. Simplify the numerator and denominator:
sqrt(0) = 0, and 4^2 - 16 = 0.
The function now becomes f(x) = 0/0.
5. Apply L'Hôpital's Rule to evaluate the limit:
To resolve the indeterminate form, we can take the derivative of the function.
f'(x) = (1 / (2 * sqrt(x - 4))) / (2x)
= 1 / (4 * x * sqrt(x - 4))
Substitute the value x = 4 into the derivative:
f'(4) = 1 / (4 * 4 * sqrt(4 - 4))
= 1 / (16 * sqrt(0))
= 1 / 0.
We still have an indeterminate form of 1/0.
6. Repeat the process:
Take the derivative of f'(x) to evaluate the limit further.
f''(x) = -3 / (8 * x^2 * sqrt(x - 4))
Substitute x = 4 into the second derivative:
f''(4) = -3 / (8 * 4^2 * sqrt(4 - 4))
= -3 / (128 * sqrt(0))
= -3 / 0.
Again, we have an indeterminate form of -3/0.
7. Repeat the process once more:
Take the derivative of f''(x).
f'''(x) = 3 / (16 * x^3 * sqrt(x - 4))
Substitute x = 4 into the third derivative:
f'''(4) = 3 / (16 * 4^3 * sqrt(4 - 4))
= 3 / (1024 * sqrt(0))
= 3 / 0.
We still end up with an indeterminate form of 3/0.
Since L'Hôpital's Rule did not resolve the indeterminate form, we need to consider an alternative approach to find the limit. However, as per the given restrictions, it is not possible to construct a function that satisfies all the conditions.