Find the limit: lim_{xrightarrowinfty}frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}

$\underset{x\to \mathrm{\infty }}{lim}\frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}$
Multiply and divided by $x^5$:
$\underset{x\to \mathrm{\infty }}{lim}\frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}=\underset{x\to \mathrm{\infty }}{lim}\frac{x^5\frac{3x^6-x^4+2x^2-3}{x^5}}{x^5\frac{4x^5+2x^3-5}{x^5}}$
Divide:
$\underset{x\to \mathrm{\infty }}{lim}\frac{x^5\frac{3x^6-x^4+2x^2-3}{x^5}}{x^5\frac{4x^5+2x^3-5}{x^5}}=\underset{x\to \mathrm{\infty }}{lim}\frac{3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5}}{4+\frac{2}{x^2}-\frac{5}{x^5}}$
The limit of the quotient is the quotient of limits:
$\underset{x\to \mathrm{\infty }}{lim}\frac{3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5}}{4+\frac{2}{x^2}-\frac{5}{x^5}}=\frac{\underset{x\to \mathrm{\infty }}{lim}(3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5})}{\underset{x\to \mathrm{\infty }}{lim}(4+\frac{2}{x^2}-\frac{5}{x^5})}$
The limit of a sum/difference is the sum/difference of limits: $\frac{\underset{x\to \mathrm{\infty }}{lim}(3x-\frac{1}{x}+\frac{2}{x^3}-\frac{3}{x^5})}{\underset{x\to \mathrm{\infty }}{lim}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{(-\underset{x\to \mathrm{\infty }}{lim}\frac{3}{x^5}+\underset{x\to \mathrm{\infty }}{lim}\frac{2}{x^3}-\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}+\underset{x\to \mathrm{\infty }}{lim}3x)}{\underset{x\to \mathrm{\infty }}{lim}(4+\frac{2}{x^2}-\frac{5}{x^5})}$
The limit of a quotient is the quotient of limits:
$\frac{(-\underset{x\to \mathrm{\infty }}{lim}\frac{3}{x^5}+\underset{x\to \mathrm{\infty }}{lim}\frac{2}{x^3}-\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}+\underset{x\to \mathrm{\infty }}{lim}3x)}{\underset{x\to \mathrm{\infty }}{lim}(4+\frac{2}{x^2}-\frac{5}{x^5})}=\frac{(-\underset{x\to \mathrm{\infty }}{lim}\frac{3}{x^5}+\underset{x\to \mathrm{\infty }}{lim}\frac{2}{x^3}-\underset{x\to \mathrm{\infty }}{lim}3x)-\frac{\underset{x\to \mathrm{\infty }}{lim}1}{\underset{x\to \mathrm{\infty }}{lim}x}}{\underset{x\to \mathrm{\infty }}{lim}(4+\frac{2}{x^2}-\frac{5}{x^5})}$
The limit of a constant is equal to the constant:

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