Guess the value of the limit (if it exists) by evaluating the function at the given numbers.  (It is suggested that you report answers accurate to at least six decimal places.)
Let

f(x)= (e^(4.7x)-e^(2.3x))/(x)

We want to find the limit

lim x is 0 f(x)= lim x is 0 (e^(4.7x)-e^(2.3x))/(x)

Start by calculating the values of the function for the inputs listed in this table.

x
0.2
0.1
0.05
0.01
0.001
0.0001
0.00001

Find the f(x) inputs in the full decimal answer

Based on the values in this table, it appears

lim x is 0 (e^(4.7x)-e^(2.3x))/(x) = 

Here are the calculated values of \( f(x) = \frac{e^{4.7x} - e^{2.3x}}{x} \) for the given inputs:

\[
\begin{aligned}
f(0.2) &= 4.879537 \\
f(0.1) &= 3.413942 \\
f(0.05) &= 2.860707 \\
f(0.01) &= 2.485547 \\
f(0.001) &= 2.408415 \\
f(0.0001) &= 2.400840 \\
f(0.00001) &= 2.400084 \\
\end{aligned}
\]

Based on these values, it appears that:

\[
\lim_{x \to 0} \frac{e^{4.7x} - e^{2.3x}}{x} \approx 2.4
\]

Would you like further details or explanations?

Here are 5 related questions:
1. How do limits relate to continuity of a function at a point?
2. What is the derivative interpretation of this limit?
3. How would L'Hopital's Rule apply to this problem?
4. Can you describe how the exponential function behaves near zero?
5. What are other methods to approximate limits besides numerical evaluation?

**Tip:** When approaching limits numerically, ensure the function's behavior near the limit is stable and consistent to avoid round-off errors.

Guess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) Let f(x) = (e^(4.7x) - e^(2.3x)) / x. We want to find the limit lim x -> 0 f(x) = lim x -> 0 (e^(4.7x) - e^(2.3x)) / x. Start by calculating the values of the function for the inputs listed in this table.

Find the limit of the function (e^(4.7x) - e^(2.3x)) / x as x approaches 0 by evaluating at specific points. Includes step-by-step calculation and interpretation using limits and calculus concepts. Suitable for advanced high school or college calculus students.

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