Find the limit $\lim_{x\to3}\left(\frac{2x^3-5x^2-2x-3}{4x^3-13x^2+4x-3}\right)$

We can factor the polynomial $4x^3-13x^2+4x-3$ 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 $-3$

Next, list all divisors of the leading coefficient $a_n$, which equals $4$

The possible roots $\pm\frac{p}{q}$ of the polynomial $4x^3-13x^2+4x-3$ will then be

Trying all possible roots, we found that $3$ is a root of the polynomial. When we evaluate it in the polynomial, it gives us $0$ as a result

Now, divide the polynomial by the root we found $\left(x-3\right)$ using synthetic division (Ruffini's rule). First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $4$ and multiply by the factor $3$. Add the result to the second coefficient and then multiply this by $3$ and so on

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $\left(x-3\right)$

We can factor the polynomial $2x^3-5x^2-2x-3$ 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 $-3$

Next, list all divisors of the leading coefficient $a_n$, which equals $2$

The possible roots $\pm\frac{p}{q}$ of the polynomial $2x^3-5x^2-2x-3$ will then be

Trying all possible roots, we found that $3$ is a root of the polynomial. When we evaluate it in the polynomial, it gives us $0$ as a result

Now, divide the polynomial by the root we found $\left(x-3\right)$ using synthetic division (Ruffini's rule). First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $2$ and multiply by the factor $3$. Add the result to the second coefficient and then multiply this by $3$ and so on

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $\left(x-3\right)$

Simplify the fraction $\frac{\left(2x^{2}+x+1\right)\left(x-3\right)}{\left(4x^{2}-x+1\right)\left(x-3\right)}$ by $x-3$

Evaluate the limit $\lim_{x\to3}\left(\frac{2x^{2}+x+1}{4x^{2}-x+1}\right)$ by replacing all occurrences of $x$ by $3$

Subtract the values $1$ and $-3$

Add the values $3$ and $1$

Calculate the power $3^{2}$

Multiply $4$ times $9$

Subtract the values $36$ and $-2$

Calculate the power $3^{2}$

Multiply $2$ times $9$

Add the values $4$ and $18$

Divide $22$ by $34$