The limit represents f'(c) for a function f and a number c. Find f and c.lim2x−6x−9x→9

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The limit represents f&#039;(c) for a function f and a number c. Find f and c.lim2x−6x−9x→9

SchepperJ · Accepted Answer

f(x)−f(c)x−c=2x−6x−9   We know that f′(c)=limx→cf(x)−f(c)x−c.So c=9 and f(x)=2x.   Comparing this we the given function, the value of c and the function f(x) can be easily seen.Result:f(x)=2xc=9

Don Sumner · Accepted Answer

Given:limx→9(2x−6x−9)We notice that the expression inside the limit resembles the derivative of a function. Specifically, the derivative of x2 is 2x2x2. limx→9(2x−6x−9)=limx→9(2x2−6x−9)Now, we can see that the expression inside the limit is the derivative of the function f(x)=x2 evaluated at c=9. Therefore, f(x)=x2 and c=9.

Vasquez · Accepted Answer

Step 1. First, let&#039;s simplify the expression inside the limit:limx→92x−6x−9Step 2. Next, we&#039;ll find the derivative of the function f(x) that represents the expression inside the limit. The derivative f′(x) will give us the slope of the tangent line to the graph of f(x) at any given point.f′(x)=ddx(2x−6x−9)Step 3. We&#039;ll find the critical points of f(x) by setting the derivative f′(x) equal to zero and solving for x:f′(x)=0Step 4. Once we have the critical points, we&#039;ll evaluate f(x) at those points to determine the corresponding values of f(c):f(c)=f(x)By following these steps, we can find the function f and the number c represented by the given limit.

nick1337 · Accepted Answer

Result: f(x)=24x+C, and c=9Solution:Using the difference of squares formula, we can rewrite x as x1/2. Therefore, the expression becomes:limx→9(2x1/2−6x−9)Now, let&#039;s simplify further. We can factor out the common factor of 2 in the numerator:limx→9(2(x1/2−3x−9))Next, let&#039;s simplify the expression within the parentheses. We can find a common denominator for the fraction:limx→9(2(x(x−9)−3x−9))Simplifying the numerator further, we have:limx→9(2(x2−9x−3x−9))Now, we can cancel out the common factor of (x−9):limx→9(2(x+3))Finally, we can evaluate the limit:limx→92(x+3)=2(9+3)=2(12)=24Therefore, f′(c)=24.To determine the function f and the number c, we need to find an antiderivative of f′(x)=24. Integrating 24 with respect to x, we get:f(x)=24x+Cwhere C is the constant of integration.Since the problem states that the limit represents f′(c), we need to find the value of c. From the limit expression, we can see that x approaches 9 as x→9. Therefore, c=9.So, the function f(x) is f(x)=24x+C, and c=9.

The limit represents f'(c) for a function f and a number c. Find f and c. PS

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