Calculus Examples

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

Step 1

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

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

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 $x^{2}$ .

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{1 + \frac{- 5 x}{x^{2}} + \frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}}}$

Step 2.1.2

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

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Step 2.1.2.1

Factor $x$ out of $- 5 x$ .

$lim_{x \to \infty} \frac{1 + \frac{x \cdot - 5}{x^{2}} + \frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to \infty} \frac{1 + \frac{x \cdot - 5}{x \cdot x} + \frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{1 + \frac{x \cdot - 5}{x \cdot x} + \frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{1 + \frac{- 5}{x} + \frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}}}$

Step 2.1.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}}}$

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.

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{- 8 x}{x^{2}}}$

Step 2.2.1.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{2 + \frac{- 8 x}{x^{2}}}$

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 $- 8 x$ .

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{2 + \frac{x \cdot - 8}{x^{2}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{2 + \frac{x \cdot - 8}{x \cdot x}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{2 + \frac{x \cdot - 8}{x \cdot x}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{2 + \frac{- 8}{x}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{1 - \frac{5}{x} + \frac{4}{x^{2}}}{2 - \frac{8}{x}}$

Step 2.3

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

$\frac{lim_{x \to \infty} 1 - \frac{5}{x} + \frac{4}{x^{2}}}{lim_{x \to \infty} 2 - \frac{8}{x}}$

Step 2.4

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

$\frac{lim_{x \to \infty} 1 - lim_{x \to \infty} \frac{5}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}{lim_{x \to \infty} 2 - \frac{8}{x}}$

Step 2.5

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

$\frac{1 - lim_{x \to \infty} \frac{5}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}{lim_{x \to \infty} 2 - \frac{8}{x}}$

Step 2.6

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

$\frac{1 - 5 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}{lim_{x \to \infty} 2 - \frac{8}{x}}$

Step 3

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

$\frac{1 - 5 \cdot 0 + lim_{x \to \infty} \frac{4}{x^{2}}}{lim_{x \to \infty} 2 - \frac{8}{x}}$

Step 4

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

$\frac{1 - 5 \cdot 0 + 4 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 2 - \frac{8}{x}}$

Step 5

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

$\frac{1 - 5 \cdot 0 + 4 \cdot 0}{lim_{x \to \infty} 2 - \frac{8}{x}}$

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 $\infty$ .

$\frac{1 - 5 \cdot 0 + 4 \cdot 0}{lim_{x \to \infty} 2 - lim_{x \to \infty} \frac{8}{x}}$

Step 6.2

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

$\frac{1 - 5 \cdot 0 + 4 \cdot 0}{2 - lim_{x \to \infty} \frac{8}{x}}$

Step 6.3

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

$\frac{1 - 5 \cdot 0 + 4 \cdot 0}{2 - 8 lim_{x \to \infty} \frac{1}{x}}$

Step 7

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

$\frac{1 - 5 \cdot 0 + 4 \cdot 0}{2 - 8 \cdot 0}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $- 5$ by $0$ .

$\frac{1 + 0 + 4 \cdot 0}{2 - 8 \cdot 0}$

Step 8.1.2

Multiply $4$ by $0$ .

$\frac{1 + 0 + 0}{2 - 8 \cdot 0}$

Step 8.1.3

Add $1 + 0$ and $0$ .

$\frac{1 + 0}{2 - 8 \cdot 0}$

Step 8.1.4

Add $1$ and $0$ .

$\frac{1}{2 - 8 \cdot 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 8$ by $0$ .

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

Step 8.2.2

Add $2$ and $0$ .

$\frac{1}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$