https://file.aiquickdraw.com/mathbot/file/1725386686316dnbxcnjj.jpg

The image appears to be a screenshot of a math problem or equation displayed on a computer screen. However, the text is not entirely clear. To assist you better, I will need to extract the mathematical content from the image.

Let me process the image and see what the equation or text says.The image shows a mathematical expression which is likely related to a calculation or evaluation. From what is visible, the expression appears to be:

\[
\lim_{{x \to \infty}} \frac{{\ln(x^3 + 7)}}{x}
\]

This is a limit problem, where \( x \) approaches infinity, and you are asked to evaluate the limit of the given function.

### Solution:

To solve this, we analyze the expression:

\[
\lim_{{x \to \infty}} \frac{{\ln(x^3 + 7)}}{x}
\]

As \( x \) approaches infinity, the expression inside the logarithm can be simplified:

\[
\ln(x^3 + 7) \approx \ln(x^3)
\]

Since \( x^3 \) dominates 7 as \( x \) becomes very large, we have:

\[
\ln(x^3) = 3\ln(x)
\]

Thus, the limit becomes:

\[
\lim_{{x \to \infty}} \frac{{3\ln(x)}}{x}
\]

Now, applying L'Hopital's Rule, which is used when both the numerator and the denominator approach infinity:

Differentiate the numerator and the denominator with respect to \( x \):

- The derivative of \( 3\ln(x) \) is \( \frac{3}{x} \).
- The derivative of \( x \) is 1.

Thus, the limit simplifies to:

\[
\lim_{{x \to \infty}} \frac{3/x}{1} = \lim_{{x \to \infty}} \frac{3}{x} = 0
\]

### Final Answer:
The limit is **0**.

Would you like more details or have any questions about this problem? Here are some related questions you might find interesting:

1. How is L'Hopital's Rule applied to different types of indeterminate forms?
2. Can you explain why \( \ln(x^3 + 7) \) is approximated as \( \ln(x^3) \) for large \( x \)?
3. What are other techniques to evaluate limits involving logarithmic functions?
4. How does the limit change if the denominator were \( x^2 \) instead of \( x \)?
5. What happens if you try to find the limit without using L'Hopital's Rule?

**Tip:** When dealing with limits involving logarithms and large values, consider simplifying the logarithmic expression before applying limit rules.

The query submitted by the user involves evaluating a mathematical limit expression as x approaches infinity.

This problem involves evaluating the limit of the logarithmic function ln(x^3 + 7) divided by x as x approaches infinity. The solution includes applying L'Hopital's Rule and simplifying the logarithmic expression for large x. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation