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# Find the limit of $\frac{3w^2+5w-2}{5w^3+4w^2+1}$ as $w$ approaches $\infty$

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## Step-by-step Solution

Problem to solve:

$\lim_{w\to\infty}\left(\frac{3w^2+5w-2}{5w^3+4w^2+1}\right)$

Specify the solving method

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We can factor the polynomial $5w^3+4w^2+1$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $1$

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Learn how to solve limits to infinity problems step by step online.

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Learn how to solve limits to infinity problems step by step online. Find the limit of (3w^2+5w-2)/(5w^3+4w^2+1) as w approaches \infty. We can factor the polynomial 5w^3+4w^2+1 using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals 1. Next, list all divisors of the leading coefficient a_n, which equals 5. The possible roots \pm\frac{p}{q} of the polynomial 5w^3+4w^2+1 will then be. Trying all possible roots, we found that -1 is a root of the polynomial. When we evaluate it in the polynomial, it gives us 0 as a result.

$\lim_{w\to\infty}\left(\frac{3w^2+5w-2}{5w^3+4w^2+1}\right)$