Complete the table and use the result to estimate the limit....

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 4} \frac{[x /(x+1)]-(4 / 5)}{x-4} $$$$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} $$

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.
$$
\lim _{x \rightarrow 4} \frac{[x /(x+1)]-(4 / 5)}{x-4}
$$$$
\begin{array}{|l|l|l|l|l|l|l|}
\hline \boldsymbol{x} & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \\
\hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\
\hline
\end{array}
$$

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Key Concepts

Limit of a Function

This is the fundamental concept in calculus that describes the behavior of a function as the input approaches a specific value. In this context, it involves analyzing the value of the function expression as x gets arbitrarily close to 4, even if the function is not defined at that exact point.

Difference Quotient

The difference quotient is the expression (f(x) - f(a))/(x - a). It forms the basis for the definition of the derivative by measuring the average rate of change of the function between two points. When x approaches a, the quotient approaches the instantaneous rate of change of the function at that point.

Derivative

The derivative represents the instantaneous rate of change of a function with respect to its variable. In the problem, the limit given is essentially the derivative of the function f(x) = x/(x+1) evaluated at x = 4. This concept is central in calculus for understanding how functions change.

Graphical Analysis

Graphical analysis involves using tools such as graphing utilities to visually examine the behavior of functions. This method can help confirm analytical results by illustrating the trend of the function near the point of interest, such as verifying the slope of the curve at x = 4.

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