Calculus Examples

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

Step 1

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

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

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 $1$ .

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

Step 2.1.1.2

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

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

Step 2.1.1.3

Cancel the common factors.

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Cancel the common factor.

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

Step 2.1.1.3.3

Rewrite the expression.

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

Step 2.1.2

Move the negative in front of the fraction.

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

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.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

Factor $x$ out of $2 x$ .

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 2$ by $0$ .

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

Step 9.1.2

Add $0$ and $0$ .

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $2$ by $0$ .

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

Step 9.2.2

Add $1 + 0$ and $0$ .

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

Step 9.2.3

Add $1$ and $0$ .

$\frac{0}{1}$

Step 9.3

Divide $0$ by $1$ .

$0$