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. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{x} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array} $$

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit.
$$
\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}
$$
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array}
$$

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Key Concepts

Numerical Estimation Using a Table of Values

Numerical estimation involves creating a table of values for small increments of x close to zero to approximate the limit. This method helps visualize the behavior of the function near the point of interest and supports theoretical conclusions about the limit.

Derivative as a Limit of a Difference Quotient

The expression (e^x - 1)/x is a difference quotient that approximates the derivative of the exponential function at a specific point (x = 0 in this case). Recognizing this as the definition of a derivative helps in understanding why the limit as x approaches zero yields the derivative of e^x at zero.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. In this context, evaluating the limit as x approaches zero involves understanding what value the function (e^x - 1)/x is approaching even if direct substitution is not possible.

Exponential Function

The exponential function, specifically e^x, is a key mathematical function noted for its growth properties and its unique characteristic where its derivative is equal to itself. This property makes it an important aspect when analyzing limits and derivatives.

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