Calculus Examples

$lim_{x \to \infty} \frac{4 x^{2} + 7 x - 5}{5 x^{2} - 6 x + 4}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{4 x^{2}}{x^{2}} + \frac{7 x}{x^{2}} + \frac{- 5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

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.

$lim_{x \to \infty} \frac{\frac{4 x^{2}}{x^{2}} + \frac{7 x}{x^{2}} + \frac{- 5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.1.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{4 + \frac{7 x}{x^{2}} + \frac{- 5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

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 $7 x$ .

$lim_{x \to \infty} \frac{4 + \frac{x \cdot 7}{x^{2}} + \frac{- 5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to \infty} \frac{4 + \frac{x \cdot 7}{x \cdot x} + \frac{- 5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{4 + \frac{x \cdot 7}{x \cdot x} + \frac{- 5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{4 + \frac{7}{x} + \frac{- 5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

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.

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{\frac{5 x^{2}}{x^{2}} + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.2.1.2

Divide $5$ by $1$ .

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{5 + \frac{- 6 x}{x^{2}} + \frac{4}{x^{2}}}$

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 $- 6 x$ .

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{5 + \frac{x \cdot - 6}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{5 + \frac{x \cdot - 6}{x \cdot x} + \frac{4}{x^{2}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{5 + \frac{x \cdot - 6}{x \cdot x} + \frac{4}{x^{2}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{5 + \frac{- 6}{x} + \frac{4}{x^{2}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{4 + \frac{7}{x} - \frac{5}{x^{2}}}{5 - \frac{6}{x} + \frac{4}{x^{2}}}$

Step 2.3

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

$\frac{lim_{x \to \infty} 4 + \frac{7}{x} - \frac{5}{x^{2}}}{lim_{x \to \infty} 5 - \frac{6}{x} + \frac{4}{x^{2}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} 4 + lim_{x \to \infty} \frac{7}{x} - lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} 5 - \frac{6}{x} + \frac{4}{x^{2}}}$

Step 2.5

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

$\frac{4 + lim_{x \to \infty} \frac{7}{x} - lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} 5 - \frac{6}{x} + \frac{4}{x^{2}}}$

Step 2.6

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

$\frac{4 + 7 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} 5 - \frac{6}{x} + \frac{4}{x^{2}}}$

Step 3

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

$\frac{4 + 7 \cdot 0 - lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} 5 - \frac{6}{x} + \frac{4}{x^{2}}}$

Step 4

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

$\frac{4 + 7 \cdot 0 - 5 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 5 - \frac{6}{x} + \frac{4}{x^{2}}}$

Step 5

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

$\frac{4 + 7 \cdot 0 - 5 \cdot 0}{lim_{x \to \infty} 5 - \frac{6}{x} + \frac{4}{x^{2}}}$

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 $\infty$ .

$\frac{4 + 7 \cdot 0 - 5 \cdot 0}{lim_{x \to \infty} 5 - lim_{x \to \infty} \frac{6}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 6.2

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

$\frac{4 + 7 \cdot 0 - 5 \cdot 0}{5 - lim_{x \to \infty} \frac{6}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 6.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 + 7 \cdot 0 - 5 \cdot 0}{5 - 6 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 7

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

$\frac{4 + 7 \cdot 0 - 5 \cdot 0}{5 - 6 \cdot 0 + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 8

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

$\frac{4 + 7 \cdot 0 - 5 \cdot 0}{5 - 6 \cdot 0 + 4 lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 9

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

$\frac{4 + 7 \cdot 0 - 5 \cdot 0}{5 - 6 \cdot 0 + 4 \cdot 0}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $7$ by $0$ .

$\frac{4 + 0 - 5 \cdot 0}{5 - 6 \cdot 0 + 4 \cdot 0}$

Step 10.1.2

Multiply $- 5$ by $0$ .

$\frac{4 + 0 + 0}{5 - 6 \cdot 0 + 4 \cdot 0}$

Step 10.1.3

Add $4 + 0$ and $0$ .

$\frac{4 + 0}{5 - 6 \cdot 0 + 4 \cdot 0}$

Step 10.1.4

Add $4$ and $0$ .

$\frac{4}{5 - 6 \cdot 0 + 4 \cdot 0}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $- 6$ by $0$ .

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

Step 10.2.2

Multiply $4$ by $0$ .

$\frac{4}{5 + 0 + 0}$

Step 10.2.3

Add $5 + 0$ and $0$ .

$\frac{4}{5 + 0}$

Step 10.2.4

Add $5$ and $0$ .

$\frac{4}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{5}$

Decimal Form:

$0.8$