To find [tex]\(\lim_{A \rightarrow 0} \frac{f(2 + A) - f(2)}{A}\)[/tex] for the function [tex]\(f(x) = x^2 + 2 \ln x\)[/tex], we can use the concept of the derivative. The expression given is the definition of the derivative of the function [tex]\(f(x)\)[/tex] at the point [tex]\(x=2\)[/tex].
To solve this step-by-step:
1. Identify the function [tex]\(f(x)\)[/tex] and the point of evaluation:
[tex]\[
f(x) = x^2 + 2 \ln x
\][/tex]
The point of evaluation is [tex]\(x = 2\)[/tex].
2. Calculate the derivative of [tex]\(f(x)\)[/tex]:
The derivative [tex]\(f'(x)\)[/tex] of [tex]\(f(x)\)[/tex] is obtained by differentiating each term of [tex]\(f(x)\)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx} (x^2) + \frac{d}{dx} (2 \ln x)
\][/tex]
[tex]\[
f'(x) = 2x + \frac{2}{x}
\][/tex]
3. Evaluate the derivative at [tex]\(x = 2\)[/tex]:
[tex]\[
f'(2) = 2(2) + \frac{2}{2}
\][/tex]
[tex]\[
f'(2) = 4 + 1
\][/tex]
[tex]\[
f'(2) = 5
\][/tex]
4. Interpret the limit:
Using the definition of the derivative, we have:
[tex]\[
\lim_{A \rightarrow 0} \frac{f(2 + A) - f(2)}{A} = f'(2)
\][/tex]
5. Conclusion:
[tex]\[
\lim_{A \rightarrow 0} \frac{f(2 + A) - f(2)}{A} = 5
\][/tex]
The correct answer is:
A. 5
