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To solve the given limit problem:

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

We can simplify the expression by dividing both the numerator and denominator by \(x^6\):

\[
\lim_{x \to \infty} \frac{10x^5 + x^4 + 31}{x^6} = \lim_{x \to \infty} \left( \frac{10x^5}{x^6} + \frac{x^4}{x^6} + \frac{31}{x^6} \right)
\]

This simplifies further as:

\[
= \lim_{x \to \infty} \left( \frac{10}{x} + \frac{1}{x^2} + \frac{31}{x^6} \right)
\]

Now, as \(x \to \infty\), each of these terms approaches 0 because any term with \(x\) in the denominator will tend to 0 as \(x\) becomes very large:

\[
\frac{10}{x} \to 0, \quad \frac{1}{x^2} \to 0, \quad \frac{31}{x^6} \to 0
\]

Thus, the limit of the whole expression is:

\[
\boxed{0}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How do limits behave when functions grow at different rates (e.g., \(x^6\) vs \(x^5\))?
2. What is L'Hôpital's Rule, and how can it be used to solve limits?
3. How can you solve limits involving infinity using asymptotic behavior?
4. What are horizontal asymptotes, and how do they relate to limits at infinity?
5. How do we handle indeterminate forms like \(\frac{\infty}{\infty}\)?

**Tip:** When dealing with limits involving infinity, it's often useful to divide by the highest power of \(x\) to simplify the expression.

Find the limit of (10x^5 + x^4 + 31) / x^6 as x approaches infinity.

Solve the limit problem (10x^5 + x^4 + 31) / x^6 as x approaches infinity using algebraic simplification. This detailed solution covers the properties of limits, and how dividing by the highest power of x helps simplify the expression. Ideal for Grades 10-12.

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