To solve this, we need to compute the values of the function given by:

$f(x) = \frac{\frac{1}{x+1} - \frac{1}{5}}{x - 4}$

Step 1: Analyze the function

This is a difference quotient, which can be simplified and evaluated for different values of $x$ near 4 to estimate the limit. First, let's simplify the numerator:

$\frac{1}{x+1} - \frac{1}{5} = \frac{5 - (x+1)}{5(x+1)} = \frac{4 - x}{5(x+1)}$

Thus, the function becomes:

$f(x) = \frac{\frac{4 - x}{5(x+1)}}{x - 4} = \frac{4 - x}{5(x+1)(x - 4)}$

This can further be simplified to:

$f(x) = -\frac{1}{5(x+1)}$

Step 2: Calculate the values of $f(x)$ for different $x$

Now we compute the values of $f(x)$ for the provided values of $x$ in the table: $x = 3.9, 3.99, 3.999, 4.001, 4.01, 4.1$.

I'll calculate these values.Here are the computed values for $f(x)$ at the given $x$ values, rounded to four decimal places:

• $f(3.9) = -0.0408$
• $f(3.99) = -0.0401$
• $f(3.999) = -0.0400$
• $f(4.001) = -0.0400$
• $f(4.01) = -0.0399$
• $f(4.1) = -0.0392$

Step 3: Estimate the limit

Looking at these values, as $x$ approaches 4 from both sides, the function seems to approach $-0.04$. Therefore, the estimated limit is:

$\lim_{x \to 4} \frac{\frac{1}{x+1} - \frac{1}{5}}{x - 4} \approx -0.0400$

Would you like any further clarifications or detailed steps?

Here are some follow-up questions:

1. How do we use limits to estimate values around a discontinuity?
2. What are difference quotients and their role in calculus?
3. Can we calculate limits of more complex functions using the same process?
4. What is the significance of simplifying the function before evaluating the limit?
5. How can graphing utilities help confirm limit estimates?

Tip: When dealing with limits, simplifying the function can often reveal the limit more clearly and make computations easier!