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Answer:
⚠️ please cannot copy like these!
< / p > < p >
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let's simplify this step-by-step.
[tex]\[
\lim_{{x \to \infty}} \frac{{3x^3 + 2x^2 + x + 1}}{{x^3 + 2x + 5}}
\][/tex]
### Step 1: Identify the Dominant Terms
[tex]- In \: the \: numerator: {\( 3x^3 \)}[/tex]
[tex]- In \: the \: denominator: {\( x^3 \)}[/tex]
### Step 2: Simplify by Dividing by the Highest Power of \( x \)
Divide each term by \( x^3 \):
[tex]\[
\frac{{3x^3/x^3 + 2x^2/x^3 + x/x^3 + 1/x^3}}{{x^3/x^3 + 2x/x^3 + 5/x^3}}
\][/tex]
Simplify:
[tex]\[
\frac{{3 + \frac{2}{x} + \frac{1}{x^2} + \frac{1}{x^3}}}{{1 + \frac{2}{x^2} + \frac{5}{x^3}}}
\][/tex]
### Step 3: Evaluate the Limit as
[tex]{\( x \to \infty \) As \\( x \) \ approaches \ infinity, \\(\frac{2}{x}\), \(\frac{1}{x^2}\), and \(\frac{1}{x^3}\) \all \approach \0:}[/tex]
[tex]\[
\lim_{{x \to \infty}} \frac{{3 + 0 + 0 + 0}}{{1 + 0 + 0}} = \frac{3}{1} = 3
\][/tex]
[tex] \: \[
\boxed{3}
\][/tex]