Calculus Examples

$lim_{x \to \infty} \frac{(x + 1) (5 x - 3)}{x^{2} + 3}$

Step 1

Simplify.

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Step 1.1

Expand $(x + 1) (5 x - 3)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$lim_{x \to \infty} \frac{x (5 x - 3) + 1 (5 x - 3)}{x^{2} + 3}$

Step 1.1.2

Apply the distributive property.

$lim_{x \to \infty} \frac{x (5 x) + x \cdot - 3 + 1 (5 x - 3)}{x^{2} + 3}$

Step 1.1.3

Apply the distributive property.

$lim_{x \to \infty} \frac{x (5 x) + x \cdot - 3 + 1 (5 x) + 1 \cdot - 3}{x^{2} + 3}$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$lim_{x \to \infty} \frac{5 x \cdot x + x \cdot - 3 + 1 (5 x) + 1 \cdot - 3}{x^{2} + 3}$

Step 1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1.2.1

Move $x$ .

$lim_{x \to \infty} \frac{5 (x \cdot x) + x \cdot - 3 + 1 (5 x) + 1 \cdot - 3}{x^{2} + 3}$

Step 1.2.1.2.2

Multiply $x$ by $x$ .

$lim_{x \to \infty} \frac{5 x^{2} + x \cdot - 3 + 1 (5 x) + 1 \cdot - 3}{x^{2} + 3}$

Step 1.2.1.3

Move $- 3$ to the left of $x$ .

$lim_{x \to \infty} \frac{5 x^{2} - 3 \cdot x + 1 (5 x) + 1 \cdot - 3}{x^{2} + 3}$

Step 1.2.1.4

Multiply $5 x$ by $1$ .

$lim_{x \to \infty} \frac{5 x^{2} - 3 x + 5 x + 1 \cdot - 3}{x^{2} + 3}$

Step 1.2.1.5

Multiply $- 3$ by $1$ .

$lim_{x \to \infty} \frac{5 x^{2} - 3 x + 5 x - 3}{x^{2} + 3}$

Step 1.2.2

Add $- 3 x$ and $5 x$ .

$lim_{x \to \infty} \frac{5 x^{2} + 2 x - 3}{x^{2} + 3}$

Step 2

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

$lim_{x \to \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{- 3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{- 3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.1.2

Divide $5$ by $1$ .

$lim_{x \to \infty} \frac{5 + \frac{2 x}{x^{2}} + \frac{- 3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.2

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

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Step 3.1.2.1

Factor $x$ out of $2 x$ .

$lim_{x \to \infty} \frac{5 + \frac{x \cdot 2}{x^{2}} + \frac{- 3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

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

$lim_{x \to \infty} \frac{5 + \frac{x \cdot 2}{x \cdot x} + \frac{- 3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{5 + \frac{x \cdot 2}{x \cdot x} + \frac{- 3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{5 + \frac{2}{x} + \frac{- 3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.1.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{5 + \frac{2}{x} - \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{5 + \frac{2}{x} - \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{3}{x^{2}}}$

Step 3.2.2

Rewrite the expression.

$lim_{x \to \infty} \frac{5 + \frac{2}{x} - \frac{3}{x^{2}}}{1 + \frac{3}{x^{2}}}$

Step 3.3

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

$\frac{lim_{x \to \infty} 5 + \frac{2}{x} - \frac{3}{x^{2}}}{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}$

Step 3.4

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

$\frac{lim_{x \to \infty} 5 + lim_{x \to \infty} \frac{2}{x} - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}$

Step 3.5

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

$\frac{5 + lim_{x \to \infty} \frac{2}{x} - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}$

Step 3.6

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

$\frac{5 + 2 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}$

Step 4

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

$\frac{5 + 2 \cdot 0 - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}$

Step 5

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

$\frac{5 + 2 \cdot 0 - 3 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}$

Step 6

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

$\frac{5 + 2 \cdot 0 - 3 \cdot 0}{lim_{x \to \infty} 1 + \frac{3}{x^{2}}}$

Step 7

Evaluate the limit.

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Step 7.1

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

$\frac{5 + 2 \cdot 0 - 3 \cdot 0}{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{3}{x^{2}}}$

Step 7.2

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

$\frac{5 + 2 \cdot 0 - 3 \cdot 0}{1 + lim_{x \to \infty} \frac{3}{x^{2}}}$

Step 7.3

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

$\frac{5 + 2 \cdot 0 - 3 \cdot 0}{1 + 3 lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 8

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

$\frac{5 + 2 \cdot 0 - 3 \cdot 0}{1 + 3 \cdot 0}$

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 $0$ .

$\frac{5 + 0 - 3 \cdot 0}{1 + 3 \cdot 0}$

Step 9.1.2

Multiply $- 3$ by $0$ .

$\frac{5 + 0 + 0}{1 + 3 \cdot 0}$

Step 9.1.3

Add $5 + 0$ and $0$ .

$\frac{5 + 0}{1 + 3 \cdot 0}$

Step 9.1.4

Add $5$ and $0$ .

$\frac{5}{1 + 3 \cdot 0}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $3$ by $0$ .

$\frac{5}{1 + 0}$

Step 9.2.2

Add $1$ and $0$ .

$\frac{5}{1}$

Step 9.3

Divide $5$ by $1$ .

$5$