Complete each table and use the results to predict the...

Complete each table and use the results to predict the indicated limit, if it exists. If $f(x)=\frac{x^{3}-2 x-4}{x-2},$ find $\lim _{x \rightarrow 2} f(x)$ $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & 9.9401 & & & 10.0601 & \end{array}$$

Complete each table and use the results to predict the indicated limit, if it exists.
If $f(x)=\frac{x^{3}-2 x-4}{x-2},$ find $\lim _{x \rightarrow 2} f(x)$
$$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & 9.9401 & & & 10.0601 & \end{array}$$

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Key Concepts

Limit

A limit describes the behavior of a function as its input approaches a particular value. It is a fundamental concept in calculus used to analyze functions near points where they may not be explicitly defined or exhibit interesting behavior, which is critical for understanding continuity and change.

Removable Discontinuity

A removable discontinuity occurs when a function is undefined at a certain point due to the cancellation of a common factor in the numerator and denominator. Even though the function is not defined at that particular point, the limit exists as the variable approaches that value, allowing the discontinuity to be 'removed' by defining the function appropriately.

Factorization

Factorization involves breaking down a polynomial into products of simpler polynomials. In the context of limits, factorization is used to simplify expressions, particularly those that form an indeterminate expression like 0/0, by canceling common factors and thereby resolving the indeterminacy.

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