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The given expression is the limit of a polynomial as \( x \to \infty \):

\[
\lim_{x \to \infty} \left( 8x^4 - 3x^3 - \frac{1}{5}x + 2 \right)
\]

### Steps to solve:

1. **Identify the highest degree term**: The highest degree term in the polynomial is \( 8x^4 \). As \( x \) approaches infinity, this term dominates all other terms because higher powers of \( x \) grow much faster than lower powers.

2. **Behavior of other terms**:
   - \( -3x^3 \) grows large but is smaller than \( x^4 \) at infinity.
   - \( -\frac{1}{5}x \) and \( 2 \) are insignificant in comparison to \( x^4 \) as \( x \to \infty \).
   
3. **Conclusion**: Since the highest degree term dominates the behavior of the polynomial as \( x \to \infty \), the limit is determined by the coefficient of \( x^4 \).

Therefore, the limit is:
\[
\lim_{x \to \infty} \left( 8x^4 - 3x^3 - \frac{1}{5}x + 2 \right) = \infty
\]

Since the leading term \( 8x^4 \) tends to infinity, the whole expression tends to infinity.

### Would you like further details or clarifications?

Here are some related questions you might find interesting:

1. How would the limit change if the leading coefficient of \( x^4 \) was negative?
2. What happens to the limit if we evaluate it as \( x \to -\infty \)?
3. Can a polynomial with a higher negative degree term ever tend to infinity as \( x \to \infty \)?
4. How does the concept of limits apply to rational functions as \( x \to \infty \)?
5. What are the general rules for determining the limit of any polynomial at infinity?

**Tip**: When dealing with limits at infinity for polynomials, always focus on the highest degree term.

lim (8x^4 - 3x^3 - 1/5x + 2) as x → ∞

Learn how to find the limit of the polynomial 8x^4 - 3x^3 - 1/5x + 2 as x tends to infinity. This problem explores the behavior of polynomials and the role of the highest degree term in determining limits. Suitable for high school students in Grades 10-12.

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