Calculus Examples

$lim_{x \to - \infty} \frac{1 - x - x^{2}}{2 x^{2} - 7}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Raise $x$ to the power of $1$ .

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

Step 2.1.1.2

Factor $x$ out of $x^{1}$ .

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

Step 2.1.1.3

Cancel the common factors.

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Cancel the common factor.

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

Step 2.1.1.3.3

Rewrite the expression.

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

Step 2.1.2

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

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Step 2.1.2.1

Cancel the common factor.

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

Step 2.1.2.2

Rewrite the expression.

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

Step 2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $2$ by $1$ .

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

Step 2.2.2

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - \frac{1}{x} - 1}{2 - \frac{7}{x^{2}}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} \frac{1}{x^{2}} - \frac{1}{x} - 1}{lim_{x \to - \infty} 2 - \frac{7}{x^{2}}}$

Step 2.4

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

$\frac{lim_{x \to - \infty} \frac{1}{x^{2}} - lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} 1}{lim_{x \to - \infty} 2 - \frac{7}{x^{2}}}$

Step 3

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

$\frac{0 - lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} 1}{lim_{x \to - \infty} 2 - \frac{7}{x^{2}}}$

Step 4

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

$\frac{0 - 0 - lim_{x \to - \infty} 1}{lim_{x \to - \infty} 2 - \frac{7}{x^{2}}}$

Step 5

Evaluate the limit.

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Step 5.1

Evaluate the limit of $1$ which is constant as $x$ approaches $- \infty$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

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

$\frac{0 - 0 - 1 \cdot 1}{2 - 7 \cdot 0}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $- 1$ by $0$ .

$\frac{0 + 0 - 1 \cdot 1}{2 - 7 \cdot 0}$

Step 7.1.2

Multiply $- 1$ by $1$ .

$\frac{0 + 0 - 1}{2 - 7 \cdot 0}$

Step 7.1.3

Add $0$ and $0$ .

$\frac{0 - 1}{2 - 7 \cdot 0}$

Step 7.1.4

Subtract $1$ from $0$ .

$\frac{- 1}{2 - 7 \cdot 0}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $- 7$ by $0$ .

$\frac{- 1}{2 + 0}$

Step 7.2.2

Add $2$ and $0$ .

$\frac{- 1}{2}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$