High School

The following limit represents the derivative of some function \( f \) at some number \( a \). Find such a function \( f \) and a number \( a \).

\[
\lim_{x \to 49} \frac{\sqrt{x} - 7}{x - 49}
\]

A. \( f(x) = x \), \( a = 0 \)
B. \( f(x) = x^2 - 49 \), \( a = 7 \)
C. \( f(x) = \sqrt{x} \), \( a = 49 \)
D. \( f(x) = 3x + 2 \), \( a = 0 \)
E. \( f(x) = x^3 \), \( a = 49 \)
F. \( f(x) = \sqrt{x} \), \( a = 7 \)
G. \( f(x) = -1 \), \( a = 49 \)

Answer :

Final answer:

The value of a is 49, and the function whose derivative is represented by the given limit is f(x) = √(49 - x^2).

Explanation:

The provided limit involves several terms and expressions, which can be analyzed step by step to determine the function and the value of a.

First, let's break down the given limit expression:

* lim (x→a) [ (49 - x) / (a - x) ] * [ (x - 7) / (x^2 + 7x + 4) ] * [ 3 / (7 + x) ]

Simplify each term separately:

  • The first term, lim (x→a) [ (49 - x) / (a - x) ], equals 1.
  • The second term, lim (x→a) [ (x - 7) / (x^2 + 7x + 4) ], also equals 1.
  • The third term, lim (x→a) [ 3 / (7 + x) ], equals 3/7.

Now, combining these simplified terms: 1 * 1 * (3/7) = 3/7.

This value represents the derivative of the function at a. To find the function itself, we need to determine a. Since the denominator in the third term involves a sum of 7 and x, and the limit expression is symmetric around a, we conclude that "a" should be 49 to make the denominators match.

Thus, the final function is: f(x) = √(49 - x^2), and the value of a is 49.

This function f(x) represents the derivative of the given limit expression with respect to x evaluated at x = 49. It is the square root of the difference between 49 and [tex]x^2[/tex], which is a common form for derivatives of inverse trigonometric functions.

Learn more about function

brainly.com/question/30721594

#SPJ11

Other Questions