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The limit problem given is:

\[
\lim_{x \to \infty} \frac{7 - 4e^x}{-5 + 3e^x}.
\]

To solve this, let's analyze the behavior of the function as \( x \to \infty \).

### Step 1: Divide by the highest exponential term
As \( x \to \infty \), the dominant term in both the numerator and the denominator is \( e^x \). To simplify, divide every term by \( e^x \):

\[
\frac{7 - 4e^x}{-5 + 3e^x} = \frac{\frac{7}{e^x} - 4}{\frac{-5}{e^x} + 3}.
\]

### Step 2: Simplify the terms
As \( x \to \infty \), \(\frac{7}{e^x} \) and \(\frac{-5}{e^x} \) approach 0, so the expression simplifies to:

\[
\frac{0 - 4}{0 + 3} = \frac{-4}{3}.
\]

### Final Answer
\[
\lim_{x \to \infty} \frac{7 - 4e^x}{-5 + 3e^x} = -\frac{4}{3}.
\]

Would you like more details or have any questions?

### Related Questions:
1. How do limits involving exponential functions behave as \( x \to -\infty \)?
2. What happens when both the numerator and the denominator grow at the same rate?
3. How do you handle limits that result in an indeterminate form like \(\frac{0}{0}\)?
4. Can this limit be evaluated using L'Hopital's Rule?
5. How do polynomial functions compare to exponential functions in terms of growth?

### Tip:
When dealing with limits at infinity, focus on the terms that grow the fastest—they often dominate the behavior of the function.

Find the following limit: lim (x -> ∞) [ (7 - 4e^x) / (-5 + 3e^x) ]. Notes: Enter 'DNE' if the limit does not exist.

Discover how to find the limit as x approaches infinity for the function (7 - 4e^x) / (-5 + 3e^x). This problem explores key calculus concepts, including the behavior of exponential functions at infinity and the application of limit laws. Suitable for students in Grades 11-12 or early college.

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