Complete the table of values (to five decimal places), and...

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}+x-6} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {1.999} & {2.001} & {2.01} & {2.1} \\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit.
$$
\lim _{x \rightarrow 2} \frac{x-2}{x^{2}+x-6}
$$
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {1.999} & {2.001} & {2.01} & {2.1} \\ \hline f(x) & {} & {} & {} \\ \hline\end{array}
$$

Key Concepts

Limit of a Function

The concept of a limit involves determining the value that a function approaches as the input approaches a specific point. This is a fundamental idea in calculus as it provides the basis for defining continuity, derivatives, and integrals.

Indeterminate Forms

An indeterminate form, such as 0/0, occurs when substituting the value into a function gives an ambiguous result. It signals that more analysis or algebraic manipulation is needed to properly evaluate the limit.

Algebraic Simplification

Algebraic techniques, such as factorization and cancellation of common terms, are often used to resolve indeterminate forms. By simplifying a function, the true behavior of the limit can be revealed without the ambiguity that a direct substitution might cause.

Numerical Estimation of Limits

Using tables of values is a practical method to estimate limits by computing function values that are increasingly close to the point of interest. This helps confirm the expected limit value, especially when direct algebraic simplification is complex or not easily apparent.

Left-hand and Right-hand Limits

The concept of left-hand and right-hand limits examines how a function behaves as it approaches a point from values less than and greater than that point. Consistency between these one-sided limits is necessary for the overall limit to exist at that point.

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