Calculus Examples

$lim_{x \to - \infty} \frac{4 x}{x^{2} - 8 x + 3}$

Step 1

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

$4 lim_{x \to - \infty} \frac{x}{x^{2} - 8 x + 3}$

Step 2

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

$4 lim_{x \to - \infty} \frac{\frac{x}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3

Evaluate the limit.

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Step 3.1

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

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Step 3.1.1

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

$4 lim_{x \to - \infty} \frac{\frac{x^{1}}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.2

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

$4 lim_{x \to - \infty} \frac{\frac{x \cdot 1}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.3

Cancel the common factors.

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Step 3.1.3.1

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

$4 lim_{x \to - \infty} \frac{\frac{x \cdot 1}{x \cdot x}}{\frac{x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.3.2

Cancel the common factor.

$4 lim_{x \to - \infty} \frac{\frac{x \cdot 1}{x \cdot x}}{\frac{x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.3.3

Rewrite the expression.

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{\frac{x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{\frac{x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.2.1.2

Rewrite the expression.

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{1 + \frac{- 8 x}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.2.2

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

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Step 3.2.2.1

Factor $x$ out of $- 8 x$ .

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{1 + \frac{x \cdot - 8}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

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

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{1 + \frac{x \cdot - 8}{x \cdot x} + \frac{3}{x^{2}}}$

Step 3.2.2.2.2

Cancel the common factor.

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{1 + \frac{x \cdot - 8}{x \cdot x} + \frac{3}{x^{2}}}$

Step 3.2.2.2.3

Rewrite the expression.

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{1 + \frac{- 8}{x} + \frac{3}{x^{2}}}$

Step 3.2.3

Move the negative in front of the fraction.

$4 lim_{x \to - \infty} \frac{\frac{1}{x}}{1 - \frac{8}{x} + \frac{3}{x^{2}}}$

Step 3.3

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

$4 \frac{lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1 - \frac{8}{x} + \frac{3}{x^{2}}}$

Step 4

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

$4 \frac{0}{lim_{x \to - \infty} 1 - \frac{8}{x} + \frac{3}{x^{2}}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$4 \frac{0}{lim_{x \to - \infty} 1 - lim_{x \to - \infty} \frac{8}{x} + lim_{x \to - \infty} \frac{3}{x^{2}}}$

Step 5.2

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

$4 \frac{0}{1 - lim_{x \to - \infty} \frac{8}{x} + lim_{x \to - \infty} \frac{3}{x^{2}}}$

Step 5.3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$4 \frac{0}{1 - 8 lim_{x \to - \infty} \frac{1}{x} + lim_{x \to - \infty} \frac{3}{x^{2}}}$

Step 6

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

$4 \frac{0}{1 - 8 \cdot 0 + lim_{x \to - \infty} \frac{3}{x^{2}}}$

Step 7

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

$4 \frac{0}{1 - 8 \cdot 0 + 3 lim_{x \to - \infty} \frac{1}{x^{2}}}$

Step 8

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

$4 \frac{0}{1 - 8 \cdot 0 + 3 \cdot 0}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Multiply $- 8$ by $0$ .

$4 \frac{0}{1 + 0 + 3 \cdot 0}$

Step 9.1.2

Multiply $3$ by $0$ .

$4 \frac{0}{1 + 0 + 0}$

Step 9.1.3

Add $1 + 0$ and $0$ .

$4 \frac{0}{1 + 0}$

Step 9.1.4

Add $1$ and $0$ .

$4 (\frac{0}{1})$

Step 9.2

Divide $0$ by $1$ .

$4 \cdot 0$

Step 9.3

Multiply $4$ by $0$ .

$0$