Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

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

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Divide $5$ by $1$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $10$ by $1$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

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

Step 3

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

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

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

$\frac{5 + 0}{10 - 3 \cdot 0 + 7 \cdot 0}$

Step 8

Simplify the answer.

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Step 8.1

Add $5$ and $0$ .

$\frac{5}{10 - 3 \cdot 0 + 7 \cdot 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 3$ by $0$ .

$\frac{5}{10 + 0 + 7 \cdot 0}$

Step 8.2.2

Multiply $7$ by $0$ .

$\frac{5}{10 + 0 + 0}$

Step 8.2.3

Add $10 + 0$ and $0$ .

$\frac{5}{10 + 0}$

Step 8.2.4

Add $10$ and $0$ .

$\frac{5}{10}$

Step 8.3

Cancel the common factor of $5$ and $10$ .

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Step 8.3.1

Factor $5$ out of $5$ .

$\frac{5 (1)}{10}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 1}{5 \cdot 2}$

Step 8.3.2.2

Cancel the common factor.

$\frac{5 \cdot 1}{5 \cdot 2}$

Step 8.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$