To find the limit of the function [tex]\(\frac{x^3 + 3x^2 + 1}{x + 2}\)[/tex] as [tex]\(x\)[/tex] approaches infinity, follow these steps:
1. Identify the degrees of the numerator and denominator:
- The numerator is [tex]\(x^3 + 3x^2 + 1\)[/tex].
- The denominator is [tex]\(x + 2\)[/tex].
- The highest degree term in the numerator is [tex]\(x^3\)[/tex].
- The highest degree term in the denominator is [tex]\(x\)[/tex].
2. Simplify the function by focusing on the terms with the highest degrees:
- When [tex]\(x\)[/tex] becomes very large (approaching infinity), the terms with the highest degrees dominate the behavior of the function.
- Hence, for large [tex]\(x\)[/tex], the function [tex]\(\frac{x^3 + 3x^2 + 1}{x + 2}\)[/tex] behaves like [tex]\(\frac{x^3}{x}\)[/tex].
3. Divide the numerator and denominator by [tex]\(x\)[/tex]:
- [tex]\[\frac{x^3 + 3x^2 + 1}{x + 2} = \frac{x^3(1 + \frac{3x^2}{x^3} + \frac{1}{x^3})}{x(1 + \frac{2}{x})}\][/tex]
- This simplifies to [tex]\[\frac{x^3(1 + \frac{3}{x} + \frac{1}{x^3})}{x(1 + \frac{2}{x})} = \frac{x^3}{x} \cdot \frac{(1 + \frac{3}{x} + \frac{1}{x^3})}{(1 + \frac{2}{x})}\][/tex]
- Simplifying further: [tex]\[\frac{x^3}{x} \cdot \frac{(1 + \frac{3}{x} + \frac{1}{x^3})}{(1 + \frac{2}{x})} = x^2 \cdot \frac{(1 + \frac{3}{x} + \frac{1}{x^3})}{(1 + \frac{2}{x})}\][/tex]
4. Consider the behavior as [tex]\(x\)[/tex] approaches infinity:
- As [tex]\(x\)[/tex] approaches infinity, [tex]\(\frac{3}{x}\)[/tex], [tex]\(\frac{1}{x^3}\)[/tex], and [tex]\(\frac{2}{x}\)[/tex] all approach 0.
- This simplifies our function to: [tex]\[x^2 \cdot \frac{1 + 0 + 0}{1 + 0} = x^2 \cdot 1 = x^2\][/tex]
5. Determine the limit:
- As [tex]\(x\)[/tex] approaches infinity, [tex]\(x^2\)[/tex] also approaches infinity.
Thus, the limit is:
[tex]\[\lim _{x \rightarrow \infty} \frac{x^3 + 3 x^2 + 1}{x + 2} = \infty.\][/tex]
