In Exercises 71 - 74 , complete the following tables and...

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$ $$f ( x ) = \frac { 10 ^ { x } - 1 } { x }$$

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = \frac { 10 ^ { x } - 1 } { x }$$

Key Concepts

Limit

A limit is the value that a function approaches as the input approaches a particular point. Here, evaluating the limit involves examining the behavior of f(x) as x gets arbitrarily close to zero from both sides.

Indeterminate Forms

When substituting the limit value directly into an expression results in an indeterminate form (like 0/0), this signals that more refined techniques must be employed to resolve the limit, such as algebraic manipulation or using the definition of a derivative.

Exponential Functions

Exponential functions, such as those with a constant base raised to a variable exponent, exhibit continuous growth or decay. Their characteristics are crucial when analyzing limits because they often lead to indeterminate forms under certain operations.

Derivative as a Limit

The definition of the derivative as a limit of a difference quotient is applicable in this context. For a function of the form (a^x - a^0)/x as x approaches zero, the limit represents the derivative of the exponential function at zero.

Logarithmic Functions in Differentiation

The derivative of an exponential function with base a involves the natural logarithm, as it allows us to relate the rate of change of the function to its current value. In particular, the derivative of a^x at zero is given by ln(a), highlighting the role of logarithms in evaluating such limits.

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