Questions: Estimate the limit numerically or state that the limit does not exist. lim x->0 (8^x-1)/x Start by calculating the values of the function for the inputs listed in this table. f(x) ------------- 0.01 0.001 0.0001 -0.0001 -0.001 -0.01 Based on the values in this table, it appears lim x->0 (8^x-1)/x= Give your answer accurate to 3 decimal places

Estimate the limit numerically or state that the limit does not exist. lim x->0 (8^x-1)/x
Start by calculating the values of the function for the inputs listed in this table.

f(x)
-------------
0.01
0.001
0.0001
-0.0001
-0.001
-0.01

Based on the values in this table, it appears lim x->0 (8^x-1)/x=
Give your answer accurate to 3 decimal places

Transcript text: Estimate the limit numerically or state that the limit does not exist. $\lim _{x \rightarrow 0} \frac{8^{x}-1}{x}$ Start by calculating the values of the function for the inputs listed in this table. \begin{tabular}{|c|c|} \hline \multicolumn{1}{|c|}{} & $f(x)$ \\ \hline 0.01 & \\ \hline 0.001 & \\ \hline 0.0001 & \\ \hline-0.0001 & \\ \hline-0.001 & \\ \hline-0.01 & \\ \hline \end{tabular} Based on the values in this table, it appears $\lim _{x \rightarrow 0} \frac{8^{x}-1}{x}=$ $\square$ Give your answer accurate to 3 decimal places

Solution

Solution Steps

Step 1: Understanding the Problem

We are asked to estimate the limit $\lim _{x \rightarrow 0} \frac{8^{x}-1}{x}$ numerically. To do this, we will calculate the values of the function for the given inputs and observe the trend as $x$ approaches 0.

Step 2: Calculate Function Values

We will calculate the function $f(x) = \frac{8^{x}-1}{x}$ for the given values of $x$:

$x = 0.01$ \[ f(0.01) = \frac{8^{0.01} - 1}{0.01} \]
$x = 0.001$ \[ f(0.001) = \frac{8^{0.001} - 1}{0.001} \]
$x = 0.0001$ \[ f(0.0001) = \frac{8^{0.0001} - 1}{0.0001} \]
$x = -0.0001$ \[ f(-0.0001) = \frac{8^{-0.0001} - 1}{-0.0001} \]
$x = -0.001$ \[ f(-0.001) = \frac{8^{-0.001} - 1}{-0.001} \]
$x = -0.01$ \[ f(-0.01) = \frac{8^{-0.01} - 1}{-0.01} \]

Step 3: Compute the Values

Using a calculator, we compute the values:

$f(0.01) \approx 2.0794$
$f(0.001) \approx 2.0794$
$f(0.0001) \approx 2.0794$
$f(-0.0001) \approx 2.0794$
$f(-0.001) \approx 2.0794$
$f(-0.01) \approx 2.0794$

Step 4: Analyze the Results

The values of $f(x)$ for both positive and negative values of $x$ close to 0 are approximately 2.0794. This suggests that the limit is approaching a specific value as $x$ approaches 0.