To evaluate the limit [tex]\(\lim_{x \rightarrow \infty} \frac{3}{x^3 + 1}\)[/tex], we can proceed step by step, considering the behavior of the function as [tex]\(x\)[/tex] approaches infinity.
1. Understand the form of the function:
The given function is [tex]\(\frac{3}{x^3 + 1}\)[/tex]. As [tex]\(x\)[/tex] increases towards infinity, the term [tex]\(x^3 + 1\)[/tex] also increases significantly.
2. Analyze the denominator:
When [tex]\(x\)[/tex] becomes very large, the expression [tex]\(x^3 + 1\)[/tex] is dominated by the [tex]\(x^3\)[/tex] term. Hence, for very large [tex]\(x\)[/tex], we can approximate [tex]\(x^3 + 1 \approx x^3\)[/tex].
3. Simplified function analysis:
Thus, the function can be approximated as [tex]\(\frac{3}{x^3 + 1} \approx \frac{3}{x^3}\)[/tex] for large values of [tex]\(x\)[/tex].
4. Calculating the limit:
Now, we analyze the limit of the simpler function,
[tex]\[
\lim_{x \rightarrow \infty} \frac{3}{x^3}
\][/tex]
As [tex]\(x\)[/tex] approaches infinity, [tex]\(x^3\)[/tex] also approaches infinity. Since [tex]\(3\)[/tex] is a constant, and the denominator [tex]\(x^3\)[/tex] grows without bound, the fraction [tex]\(\frac{3}{x^3}\)[/tex] becomes very small. Specifically, it approaches zero.
5. Conclusion:
Hence, the value of the limit is:
[tex]\[
\lim_{x \rightarrow \infty} \frac{3}{x^3 + 1} = 0
\][/tex]
Therefore, the limit is [tex]\(0\)[/tex].
