Let's start by analyzing the function and completing the steps outlined in the question.

Step 1: Function Definition and Domain

The given function is: $f(x) = \frac{-1 + \sqrt{1+x}}{x}$ We are tasked with determining values of $f(x)$ for specific $x$-values, estimating a limiting value as $x \to 0$, and finally, rationalizing the function to find a continuous extension.

Step 2: Table of Values for $f(x)$

We will evaluate $f(x)$ at the following values: $x = \pm 1, \pm 0.1, \pm 0.01, \pm 0.001$.

I’ll compute these values next.Here's the table of values for $f(x)$ at the specified points:

\hline x & f(x) \\ \hline -1 & 1.0000 \\ -0.1 & 0.5132 \\ -0.01 & 0.5013 \\ -0.001 & 0.5001 \\ 0.001 & 0.4999 \\ 0.01 & 0.4988 \\ 0.1 & 0.4881 \\ 1 & 0.4142 \\ \hline \end{array}$$ ### Step 3: Guessing the Limiting Value As $$x$$ approaches 0 from both positive and negative sides, the values of $$f(x)$$ seem to approach 0.5. Thus, we can hypothesize that: $$\lim_{x \to 0} f(x) = 0.5$$ ### Step 4: Rationalization and Continuous Function $$g(x)$$ To find a continuous function $$g(x)$$, we rationalize the expression for $$f(x)$$. Starting from: $$f(x) = \frac{-1 + \sqrt{1+x}}{x}$$ Multiply both the numerator and the denominator by the conjugate of the numerator, which is $$-1 - \sqrt{1+x}$$, to simplify. Let's compute this rationalization next.It seems like I can’t do more advanced data analysis right now. Please try again later. I’ll explain the rationalization step verbally for now. ### Rationalization of $$f(x)$$: We begin with: $$f(x) = \frac{-1 + \sqrt{1+x}}{x}$$ Multiply both the numerator and denominator by the conjugate of the numerator: $$\frac{-1 + \sqrt{1+x}}{x} \times \frac{-1 - \sqrt{1+x}}{-1 - \sqrt{1+x}} = \frac{(-1 + \sqrt{1+x})(-1 - \sqrt{1+x})}{x(-1 - \sqrt{1+x})}$$ The numerator becomes: $$(-1 + \sqrt{1+x})(-1 - \sqrt{1+x}) = -(1+x) + 1 = -x$$ Thus, the expression simplifies to: $$f(x) = \frac{-x}{x(-1 - \sqrt{1+x})} = \frac{1}{1 + \sqrt{1+x}}$$ This new function is: $$g(x) = \frac{1}{1 + \sqrt{1+x}}$$ ### Step 5: Proving the Limit is 0.5 As $$x \to 0$$, the expression $$g(x) = \frac{1}{1 + \sqrt{1+x}}$$ approaches: \[ g(0) = \frac{1}{1 + \sqrt{1+