limit of x approaching infinity 3 divided by "e" to the "x" power minus eight

The given limit is:

\[
\lim_{{x \to \infty}} \frac{3}{{e^x - 8}}
\]

To solve this, let's examine the behavior of the function as \( x \) approaches infinity.

1. The exponential function \( e^x \) grows very quickly as \( x \to \infty \). Therefore, \( e^x - 8 \) will also grow without bound.
2. Since the denominator \( e^x - 8 \) grows much faster than the constant numerator \( 3 \), the overall value of the fraction approaches 0.

Thus, the limit is:

\[
\lim_{{x \to \infty}} \frac{3}{{e^x - 8}} = 0
\]

This result occurs because the denominator becomes infinitely large, causing the fraction to shrink towards 0.

Would you like further clarification or details?

Here are 5 related questions to expand on this concept:
1. What is the behavior of \( \frac{1}{{e^x}} \) as \( x \) approaches negative infinity?
2. How does the growth of exponential functions compare to polynomial functions?
3. Can you find the limit of \( \frac{3}{{e^x}} \) as \( x \to 0 \)?
4. What is the limit of \( e^{-x} \) as \( x \to \infty \)?
5. How does \( e^x \) behave for values of \( x \) close to 0?

**Tip:** Exponential functions grow or decay much faster than polynomial functions as \( x \to \infty \).

limit of x approaching infinity 3 divided by 'e' to the 'x' power minus eight

Understand how to calculate the limit of 3 divided by e^x - 8 as x approaches infinity. This solution covers concepts like limits, exponential growth, and asymptotic behavior, suitable for students studying calculus in Grades 11-12.

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