Create a table of values for the function and use the result...

Create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.
$$\lim _{x \rightarrow 2} \frac{[x /(x+1)]-(2 / 3)}{x-2}$$

Key Concepts

Graphical Analysis

Graphical analysis uses visual representations to confirm the behavior of a function near a point of interest. By plotting the function and examining its behavior as the input approaches the limit point, one can visually verify the limit computed numerically or analytically.

Numerical Estimation using Tables

Creating a table of values is a numerical estimation method used in calculus to observe the behavior of a function as the input variable gets arbitrarily close to a specific point. This technique helps in approximating the limit by evaluating the function at values near the target value.

Indeterminate Form

When evaluating a limit, the expression may take on forms such as 0/0 which do not provide immediate information about the limit's value; these are termed indeterminate forms. Recognizing an indeterminate form is crucial because it signals the need for further analysis or alternative techniques (like algebraic simplification or using derivatives) to find the limit.

Limit

A limit describes the behavior of a function as its input approaches a specific value. In the context of calculus, it provides a method for defining concepts such as continuity and the derivative, even when the function is not directly defined or is indeterminate at that point.

Difference Quotient and Derivative

The expression given in the limit resembles the difference quotient, which is used to define the derivative of a function at a point. The derivative represents the instantaneous rate of change of the function and can be calculated as the limit of the difference quotient as the input approaches the point of interest.

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