To find the limit of the expression [tex]\( x^5 - 1000x^4 \)[/tex] as [tex]\( x \)[/tex] approaches infinity, let's analyze how both terms behave as [tex]\( x \)[/tex] becomes very large.
1. Identify the leading terms:
- The given expression is [tex]\( x^5 - 1000x^4 \)[/tex].
- Here, [tex]\( x^5 \)[/tex] and [tex]\( 1000x^4 \)[/tex] are the two terms of interest.
- As [tex]\( x \)[/tex] becomes very large, the higher power term [tex]\( x^5 \)[/tex] will dominate the behavior of the expression.
2. Analyze the terms separately:
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( x^5 \)[/tex] increases much faster than [tex]\( 1000x^4 \)[/tex].
- Let's note the behavior of each term:
- [tex]\( x^5 \)[/tex] approaches infinity ([tex]\( \infty \)[/tex]) as [tex]\( x \)[/tex] approaches infinity.
- [tex]\( 1000x^4 \)[/tex] also approaches infinity ([tex]\( \infty \)[/tex]) as [tex]\( x \)[/tex] approaches infinity, but at a slower rate compared to [tex]\( x^5 \)[/tex].
3. Compare the leading terms:
- Since [tex]\( x^5 \)[/tex] grows significantly faster than [tex]\( 1000x^4 \)[/tex], the term [tex]\( -1000x^4 \)[/tex] becomes negligible relative to [tex]\( x^5 \)[/tex] for very large [tex]\( x \)[/tex].
- Therefore, the dominant term [tex]\( x^5 \)[/tex] dictates the behavior of the entire expression as [tex]\( x \)[/tex] approaches infinity.
4. Conclusion:
- Given the dominance of the [tex]\( x^5 \)[/tex] term, the limit of the expression [tex]\( x^5 - 1000x^4 \)[/tex] as [tex]\( x \)[/tex] approaches infinity is essentially equivalent to the limit of [tex]\( x^5 \)[/tex] as [tex]\( x \)[/tex] approaches infinity.
Hence, the limit is:
[tex]\[
\lim_{x \to \infty} \left( x^5 - 1000x^4 \right) = \infty
\][/tex]
