Calculus Examples

$lim_{x \to - \infty} \frac{2 x^{4} - 8 x + x + 2}{4 x^{4} - 5 x^{2} - 2}$

Step 1

Add $- 8 x$ and $x$ .

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

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{4}$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

Cancel the common factor of $x^{4}$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Divide $2$ by $1$ .

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

Step 3.1.2

Cancel the common factor of $x$ and $x^{4}$ .

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Step 3.1.2.1

Factor $x$ out of $- 7 x$ .

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

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

Factor $x$ out of $x^{4}$ .

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

Step 3.1.2.2.2

Cancel the common factor.

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

Step 3.1.2.2.3

Rewrite the expression.

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

Step 3.1.3

Move the negative in front of the fraction.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $x^{4}$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $4$ by $1$ .

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

Step 3.2.2

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 3.2.2.1

Factor $x^{2}$ out of $- 5 x^{2}$ .

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

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 3.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.3

Rewrite the expression.

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 3.2.4

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Evaluate the limit of $2$ which is constant as $x$ approaches $- \infty$ .

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

Step 3.6

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

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

Step 4

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

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

Step 5

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

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

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

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

Step 7

Evaluate the limit.

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Step 7.1

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

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

Step 7.2

Evaluate the limit of $4$ which is constant as $x$ approaches $- \infty$ .

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

Step 7.3

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

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

Step 8

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

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

Step 9

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

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

Step 10

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $- 7$ by $0$ .

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

Step 11.1.2

Multiply $2$ by $0$ .

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

Step 11.1.3

Add $2 + 0$ and $0$ .

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

Step 11.1.4

Add $2$ and $0$ .

$\frac{2}{4 - 5 \cdot 0 - 2 \cdot 0}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $- 5$ by $0$ .

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

Step 11.2.2

Multiply $- 2$ by $0$ .

$\frac{2}{4 + 0 + 0}$

Step 11.2.3

Add $4 + 0$ and $0$ .

$\frac{2}{4 + 0}$

Step 11.2.4

Add $4$ and $0$ .

$\frac{2}{4}$

Step 11.3

Cancel the common factor of $2$ and $4$ .

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Step 11.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $2$ out of $4$ .

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

Step 11.3.2.2

Cancel the common factor.

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

Step 11.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$