- Evaluate the limit for \(b\) by rationalizing the numerator to get \(b = \frac{1}{2}\).
- Infer the function \(g(x) = 4x - x^2\) from the definition of its derivative at \(x=2\).
- Calculate \(g(b)\) by substituting \(b = \frac{1}{2}\) into \(g(x)\) to obtain \(\frac{7}{4}\).
- The final answer is \(\boxed{\frac{7}{4}}\).
Explanation
- Evaluating the limit for b
To determine the value of \(g(b)\), we first need to evaluate the limit defined for \(b\). The expression is given as \(b = \lim _{x \rightarrow -3} \frac{\sqrt{x+4}-1}{x+3}\). As \(x \to -3\), the numerator becomes \(\sqrt{-3+4}-1 = 1-1 = 0\) and the denominator becomes \(-3+3 = 0\), indicating an indeterminate form of \(\frac{0}{0}\). We can resolve this by rationalizing the numerator:
\[b = \lim _{x \rightarrow -3} \frac{\sqrt{x+4}-1}{x+3} \cdot \frac{\sqrt{x+4}+1}{\sqrt{x+4}+1} = \lim _{x \rightarrow -3} \frac{(x+4)-1}{(x+3)(\sqrt{x+4}+1)}\]
Simplifying the expression, we get:
\[b = \lim _{x \rightarrow -3} \frac{x+3}{(x+3)(\sqrt{x+4}+1)} = \lim _{x \rightarrow -3} \frac{1}{\sqrt{x+4}+1}\]
Evaluating this at \(x = -3\):
\[b = \frac{1}{\sqrt{-3+4}+1} = \frac{1}{1+1} = \frac{1}{2}\]
Thus, the value of \(b\) is \(\frac{1}{2}\).
- Identifying the function g(x)
Next, we examine the function \(g(x)\). The problem provides the derivative at a point: \(g'(2) = \lim _{x \rightarrow 2} \frac{(4x-x^2)-4}{x-2}\). Evaluating this limit:
\[g'(2) = \lim _{x \rightarrow 2} \frac{-(x^2-4x+4)}{x-2} = \lim _{x \rightarrow 2} \frac{-(x-2)^2}{x-2} = \lim _{x \rightarrow 2} -(x-2) = 0\]
Given the structure of the expression \((4x-x^2)\) within the limit for the derivative, it is implied that the function \(g(x)\) is defined as \(g(x) = 4x - x^2\). We can verify this: if \(g(x) = 4x - x^2\), then the derivative is \(g'(x) = 4 - 2x\), which evaluated at \(x=2\) gives \(g'(2) = 4 - 2(2) = 0\). This confirms the consistency of the assumption.
- Calculating g(b) and Final Answer
Finally, we evaluate \(g(b)\) using the function \(g(x) = 4x - x^2\) and the value \(b = \frac{1}{2}\):
\[g(b) = g\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right) - \left(\frac{1}{2}\right)^2\]
Calculating this:
\[g\left(\frac{1}{2}\right) = 2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4}\]
Comparing this to the given options, the result corresponds to option (C). Therefore, the value of \(g(b)\) is \(\frac{7}{4}\).
Examples
Understanding limits and functional relationships is fundamental in fields like physics and engineering, where we often need to analyze the behavior of systems near critical points. For instance, calculating the instantaneous velocity or the rate of change of a system defined by complex curves allows engineers to ensure stability and precision in design. This problem demonstrates how evaluating limits helps identify the underlying functional relationships, which is a key skill for modeling dynamic processes in the real world.