Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Rewrite using the commutative property of multiplication.

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

Step 1.3

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

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

Step 1.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.4.1

Move $x^{2}$ .

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

Step 1.4.2

Multiply $x^{2}$ by $x$ .

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Step 1.4.2.1

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

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

Step 1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.4.3

Add $2$ and $1$ .

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

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

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.1.1

Multiply by $1$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $x^{3}$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $2$ by $1$ .

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

Step 3.2.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 3.2.2.1

Factor $x$ out of $3 x$ .

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

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $x$ out of $x^{3}$ .

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

Step 3.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

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

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

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

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

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 $\infty$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $5$ by $0$ .

$\frac{0 + 0}{2 + 3 \cdot 0}$

Step 9.1.2

Add $0$ and $0$ .

$\frac{0}{2 + 3 \cdot 0}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $3$ by $0$ .

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

Step 9.2.2

Add $2$ and $0$ .

$\frac{0}{2}$

Step 9.3

Divide $0$ by $2$ .

$0$