limit of 3 times 0.74 ^ x as X approaches negative infinity

We are tasked with finding the limit of the expression \( 3 \times 0.74^x \) as \( x \) approaches negative infinity:

\[
\lim_{x \to -\infty} 3 \times 0.74^x
\]

### Step-by-Step Solution:

1. The constant \( 3 \) does not affect the limit behavior of \( 0.74^x \), so we can focus on \( 0.74^x \).

2. Since \( 0.74 \) is a number between 0 and 1, raising it to a large positive power results in smaller and smaller values. Conversely, when raised to increasingly negative powers, it behaves like:

\[
0.74^x = \frac{1}{(0.74)^{-x}}
\]

As \( x \to -\infty \), the exponent \( -x \) grows, making \( 0.74^x \to \infty \). Therefore, the expression grows unbounded.

3. Hence, the limit becomes:

\[
\lim_{x \to -\infty} 3 \times 0.74^x = \infty
\]

### Final Answer:
\[
\lim_{x \to -\infty} 3 \times 0.74^x = \infty
\]

Would you like more details on this, or do you have any further questions?

Here are 5 related questions:
1. What is the behavior of \( 3 \times 0.74^x \) as \( x \to \infty \)?
2. How does the base affect the growth of exponential functions?
3. What happens if the base is greater than 1 as \( x \to -\infty \)?
4. Can you compare the limits of \( 0.74^x \) and \( 0.5^x \) as \( x \to -\infty \)?
5. How would the limit change if the base was negative, like \( -0.74^x \)?

**Tip:** For exponential functions with bases between 0 and 1, raising them to large negative powers leads to exponential growth.

Find the limit of 3 times 0.74 raised to the power of x as x approaches negative infinity. Understand the behavior of exponential functions with bases between 0 and 1, leading to exponential growth as x tends to negative infinity. Suitable for high school students in Grades 11-12.

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