## Final Answer

## Step-by-step Solution

Problem to solve:

Specify the solving method

As a variable goes to infinity, the expression $2x^3-2x^2+x-3$ will behave the same way that it's largest power behaves

As a variable goes to infinity, the expression $x^3+2x^2-x+1$ will behave the same way that it's largest power behaves

Plug in the value $\infty $ into the limit

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

Any expression multiplied by infinity tends to infinity

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

If we directly evaluate the limit $\lim_{x\to \infty }\left(\frac{2x^3-2x^2+x-3}{x^3+2x^2-x+1}\right)$ as $x$ tends to $\infty $, we can see that it gives us an indeterminate form

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

Find the derivative of the numerator

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($-3$) is equal to zero

The derivative of the linear function is equal to $1$

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

The derivative of a function multiplied by a constant ($-2$) is equal to the constant times the derivative of the function

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

Find the derivative of the denominator

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($1$) is equal to zero

The derivative of the linear function times a constant, is equal to the constant

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

After deriving both the numerator and denominator, the limit results in

As a variable goes to infinity, the expression $6x^{2}-4x+1$ will behave the same way that it's largest power behaves

As a variable goes to infinity, the expression $3x^{2}+4x-1$ will behave the same way that it's largest power behaves

Plug in the value $\infty $ into the limit

Infinity to the power of any positive number is equal to infinity, so $\infty ^{2}=\infty$

Any expression multiplied by infinity tends to infinity

Infinity to the power of any positive number is equal to infinity, so $\infty ^{2}=\infty$

Any expression multiplied by infinity tends to infinity

If we directly evaluate the limit $\lim_{x\to \infty }\left(\frac{6x^{2}-4x+1}{3x^{2}+4x-1}\right)$ as $x$ tends to $\infty $, we can see that it gives us an indeterminate form

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

Find the derivative of the numerator

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($1$) is equal to zero

The derivative of the linear function times a constant, is equal to the constant

The derivative of a function multiplied by a constant ($6$) is equal to the constant times the derivative of the function

Find the derivative of the denominator

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($-1$) is equal to zero

The derivative of the linear function times a constant, is equal to the constant

The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function

Factor the numerator by $2$

Factor the denominator by $2$

Cancel the fraction's common factor $2$

After deriving both the numerator and denominator, the limit results in

Plug in the value $\infty $ into the limit

Any expression multiplied by infinity tends to infinity

Infinity plus any algebraic expression is equal to infinity

Any expression multiplied by infinity tends to infinity

Infinity plus any algebraic expression is equal to infinity

If we directly evaluate the limit $\lim_{x\to \infty }\left(\frac{6x-2}{3x+2}\right)$ as $x$ tends to $\infty $, we can see that it gives us an indeterminate form

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

Find the derivative of the numerator

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($-2$) is equal to zero

The derivative of the linear function times a constant, is equal to the constant

Find the derivative of the denominator

The derivative of a sum of two or more functions is the sum of the derivatives of each function

The derivative of the constant function ($2$) is equal to zero

The derivative of the linear function times a constant, is equal to the constant

Divide $6$ by $3$

After deriving both the numerator and denominator, the limit results in

The limit of a constant is just the constant