https://file.aiquickdraw.com/mathbot/file/1725505166216ad5izglt.jpg

To solve the given limit:

\[
\lim_{{x \to \infty}} \frac{10x^5 + x^4 + 31}{x^6}
\]

We can proceed by dividing both the numerator and the denominator by the highest power of \( x \) in the denominator, which is \( x^6 \).

### Step-by-Step Solution:

1. Divide every term in the numerator by \( x^6 \):
   \[
   \frac{10x^5}{x^6} + \frac{x^4}{x^6} + \frac{31}{x^6} = \frac{10}{x} + \frac{1}{x^2} + \frac{31}{x^6}
   \]

2. Now take the limit of each term as \( x \to \infty \):
   \[
   \lim_{{x \to \infty}} \left( \frac{10}{x} + \frac{1}{x^2} + \frac{31}{x^6} \right)
   \]

   As \( x \) approaches infinity, the terms \( \frac{10}{x} \), \( \frac{1}{x^2} \), and \( \frac{31}{x^6} \) all approach 0.

3. Therefore, the limit is:
   \[
   0 + 0 + 0 = 0
   \]

### Final Answer:
\[
\lim_{{x \to \infty}} \frac{10x^5 + x^4 + 31}{x^6} = 0
\]

Let me know if you would like more details or have any further questions!

---

Here are 5 related questions for further practice:

1. What is the limit of \(\lim_{{x \to \infty}} \frac{5x^3 + 7x + 9}{x^4}\)?
2. How would the limit change if the highest power in the numerator was \(x^7\) instead of \(x^5\)?
3. How do you handle limits of rational functions when the degrees of the numerator and denominator are the same?
4. What happens to the limit of a function if the degree of the numerator is greater than that of the denominator?
5. How does L'Hopital's Rule apply to limits of rational functions approaching infinity?

### Tip:
Always simplify limits by dividing all terms by the highest power of \(x\) in the denominator to make the limit clearer when \(x\) approaches infinity.

Learn how to compute the limit of the rational function (10x^5 + x^4 + 31) / x^6 as x approaches infinity. This step-by-step solution covers key calculus concepts, including dividing by the highest power of x and applying limit theorems. Suitable for 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