Evaluate limit as x approaches infinity of (10x^3-6x^2-5x)/(4-2x-7x^3)

$lim_{x \to \infty} \frac{10 x^{3} - 6 x^{2} - 5 x}{4 - 2 x - 7 x^{3}}$

Take the limit of each term.

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Reorder $4$ and $- 2 x$ .

$lim_{x \to \infty} \frac{10 x^{3} - 6 x^{2} - 5 x}{- 2 x + 4 - 7 x^{3}}$

Move $4$ .

$lim_{x \to \infty} \frac{10 x^{3} - 6 x^{2} - 5 x}{- 2 x - 7 x^{3} + 4}$

Reorder $- 2 x$ and $- 7 x^{3}$ .

$lim_{x \to \infty} \frac{10 x^{3} - 6 x^{2} - 5 x}{- 7 x^{3} - 2 x + 4}$

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

$lim_{x \to \infty} \frac{\frac{10 x^{3}}{x^{3}} + \frac{- 6 x^{2}}{x^{3}} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Simplify each term.

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Cancel the common factor of $x^{3}$ .

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Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{10 x^{3}}{x^{3}} + \frac{- 6 x^{2}}{x^{3}} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Divide $10$ by $1$ .

$lim_{x \to \infty} \frac{10 + \frac{- 6 x^{2}}{x^{3}} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Factor $x^{2}$ out of $- 6 x^{2}$ .

$lim_{x \to \infty} \frac{10 + \frac{x^{2} \cdot - 6}{x^{3}} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factors.

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Factor $x^{2}$ out of $x^{3}$ .

$lim_{x \to \infty} \frac{10 + \frac{x^{2} \cdot - 6}{x^{2} x} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factor.

$lim_{x \to \infty} \frac{10 + \frac{x^{2} \cdot - 6}{x^{2} x} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Rewrite the expression.

$lim_{x \to \infty} \frac{10 + \frac{- 6}{x} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} + \frac{- 5 x}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factor of $x$ and $x^{3}$ .

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Factor $x$ out of $- 5 x$ .

$lim_{x \to \infty} \frac{10 - \frac{6}{x} + \frac{x \cdot - 5}{x^{3}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factors.

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Factor $x$ out of $x^{3}$ .

$lim_{x \to \infty} \frac{10 - \frac{6}{x} + \frac{x \cdot - 5}{x \cdot x^{2}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factor.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} + \frac{x \cdot - 5}{x \cdot x^{2}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Rewrite the expression.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} + \frac{- 5}{x^{2}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Simplify each term.

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Cancel the common factor of $x^{3}$ .

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Cancel the common factor.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{\frac{- 7 x^{3}}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Divide $- 7$ by $1$ .

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{- 7 + \frac{- 2 x}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factor of $x$ and $x^{3}$ .

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Factor $x$ out of $- 2 x$ .

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{- 7 + \frac{x \cdot - 2}{x^{3}} + \frac{4}{x^{3}}}$

Cancel the common factors.

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Factor $x$ out of $x^{3}$ .

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{- 7 + \frac{x \cdot - 2}{x \cdot x^{2}} + \frac{4}{x^{3}}}$

Cancel the common factor.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{- 7 + \frac{x \cdot - 2}{x \cdot x^{2}} + \frac{4}{x^{3}}}$

Rewrite the expression.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{- 7 + \frac{- 2}{x^{2}} + \frac{4}{x^{3}}}$

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{10 - \frac{6}{x} - \frac{5}{x^{2}}}{- 7 - \frac{2}{x^{2}} + \frac{4}{x^{3}}}$

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 10 - \frac{6}{x} - \frac{5}{x^{2}}}{lim_{x \to \infty} - 7 - \frac{2}{x^{2}} + \frac{4}{x^{3}}}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 10 - lim_{x \to \infty} \frac{6}{x} - lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} - 7 - \frac{2}{x^{2}} + \frac{4}{x^{3}}}$

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to \infty} 10 - 6 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} - 7 - \frac{2}{x^{2}} + \frac{4}{x^{3}}}$

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} - 7 - \frac{2}{x^{2}} + \frac{4}{x^{3}}}$

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - 5 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} - 7 - \frac{2}{x^{2}} + \frac{4}{x^{3}}}$

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - 5 \cdot 0}{lim_{x \to \infty} - 7 - \frac{2}{x^{2}} + \frac{4}{x^{3}}}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - 5 \cdot 0}{- lim_{x \to \infty} 7 - lim_{x \to \infty} \frac{2}{x^{2}} + lim_{x \to \infty} \frac{4}{x^{3}}}$

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - 5 \cdot 0}{- lim_{x \to \infty} 7 - 2 lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} \frac{4}{x^{3}}}$

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - 5 \cdot 0}{- lim_{x \to \infty} 7 - 2 \cdot 0 + lim_{x \to \infty} \frac{4}{x^{3}}}$

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - 5 \cdot 0}{- lim_{x \to \infty} 7 - 2 \cdot 0 + 4 lim_{x \to \infty} \frac{1}{x^{3}}}$

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{lim_{x \to \infty} 10 - 6 \cdot 0 - 5 \cdot 0}{- lim_{x \to \infty} 7 - 2 \cdot 0 + 4 \cdot 0}$

Evaluate the limits by plugging in the value for the variable.

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Evaluate the limit of $10$ which is constant as $x$ approaches $\infty$ .

$\frac{10 - 6 \cdot 0 - 5 \cdot 0}{- lim_{x \to \infty} 7 - 2 \cdot 0 + 4 \cdot 0}$

Evaluate the limit of $7$ which is constant as $x$ approaches $\infty$ .

$\frac{10 - 6 \cdot 0 - 5 \cdot 0}{- 7 - 2 \cdot 0 + 4 \cdot 0}$

Simplify the answer.

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Simplify the numerator.

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Multiply $- 6$ by $0$ .

$\frac{10 + 0 - 5 \cdot 0}{- 7 - 2 \cdot 0 + 4 \cdot 0}$

Multiply $- 5$ by $0$ .

$\frac{10 + 0 + 0}{- 7 - 2 \cdot 0 + 4 \cdot 0}$

Add $10 + 0$ and $0$ .

$\frac{10 + 0}{- 7 - 2 \cdot 0 + 4 \cdot 0}$

Add $10$ and $0$ .

$\frac{10}{- 7 - 2 \cdot 0 + 4 \cdot 0}$

Simplify the denominator.

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Multiply $- 2$ by $0$ .

$\frac{10}{- 7 + 0 + 4 \cdot 0}$

Multiply $4$ by $0$ .

$\frac{10}{- 7 + 0 + 0}$

Add $- 7 + 0$ and $0$ .

$\frac{10}{- 7 + 0}$

Add $- 7$ and $0$ .

$\frac{10}{- 7}$

Move the negative in front of the fraction.

$- \frac{10}{7}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{10}{7}$

Decimal Form:

$- 1.\overline{428571}$

Evaluate limit as x approaches infinity of (10x^3-6x^2-5x)/(4-2x-7x^3)

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