Calculus Examples

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

Step 1

Simplify the limit argument.

Tap for more steps...

Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Combine terms.

Tap for more steps...

Step 1.2.1

To write $5 x^{2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the limit argument.

Tap for more steps...

Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Combine factors.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Add $1$ and $2$ .

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

Step 2.2.4

Multiply $\frac{5 x^{3} + 1}{x}$ by $\frac{1}{2 x^{3} + 5}$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Move $5$ to the left of $x$ .

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

Step 3.4

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 3.4.1

Move $x^{3}$ .

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

Step 3.4.2

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.3

Add $3$ and $1$ .

$lim_{x \to \infty} \frac{5 x^{3} + 1}{2 x^{4} + 5 \cdot x}$

$lim_{x \to \infty} \frac{5 x^{3} + 1}{2 x^{4} + 5 x}$

Step 4

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{5 x^{3}}{x^{4}} + \frac{1}{x^{4}}}{\frac{2 x^{4}}{x^{4}} + \frac{5 x}{x^{4}}}$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

Cancel the common factor of $x^{3}$ and $x^{4}$ .

Tap for more steps...

Step 5.1.1

Factor $x^{3}$ out of $5 x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 5}{x^{4}} + \frac{1}{x^{4}}}{\frac{2 x^{4}}{x^{4}} + \frac{5 x}{x^{4}}}$

Step 5.1.2

Cancel the common factors.

Tap for more steps...

Step 5.1.2.1

Factor $x^{3}$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 5}{x^{3} x} + \frac{1}{x^{4}}}{\frac{2 x^{4}}{x^{4}} + \frac{5 x}{x^{4}}}$

Step 5.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 5}{x^{3} x} + \frac{1}{x^{4}}}{\frac{2 x^{4}}{x^{4}} + \frac{5 x}{x^{4}}}$

Step 5.1.2.3

Rewrite the expression.

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

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $x^{4}$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $2$ by $1$ .

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

Step 5.2.2

Cancel the common factor of $x$ and $x^{4}$ .

Tap for more steps...

Step 5.2.2.1

Factor $x$ out of $5 x$ .

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

Step 5.2.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.2.1

Factor $x$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{5}{x} + \frac{1}{x^{4}}}{2 + \frac{x \cdot 5}{x \cdot x^{3}}}$

Step 5.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x} + \frac{1}{x^{4}}}{2 + \frac{x \cdot 5}{x \cdot x^{3}}}$

Step 5.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{5}{x} + \frac{1}{x^{4}}}{2 + \frac{5}{x^{3}}}$

Step 5.3

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

$\frac{lim_{x \to \infty} \frac{5}{x} + \frac{1}{x^{4}}}{lim_{x \to \infty} 2 + \frac{5}{x^{3}}}$

Step 5.4

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

$\frac{lim_{x \to \infty} \frac{5}{x} + lim_{x \to \infty} \frac{1}{x^{4}}}{lim_{x \to \infty} 2 + \frac{5}{x^{3}}}$

Step 5.5

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

$\frac{5 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{1}{x^{4}}}{lim_{x \to \infty} 2 + \frac{5}{x^{3}}}$

Step 6

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

$\frac{5 \cdot 0 + lim_{x \to \infty} \frac{1}{x^{4}}}{lim_{x \to \infty} 2 + \frac{5}{x^{3}}}$

Step 7

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

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

Step 8

Evaluate the limit.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Cancel the common factor of $5 \cdot 0 + 0$ and $2 + 5 \cdot 0$ .

Tap for more steps...

Step 10.1.1

Reorder terms.

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

Step 10.1.2

Factor $2$ out of $5 \cdot 0$ .

$\frac{2 (5 \cdot 0) + 0}{2 + 0 \cdot 5}$

Step 10.1.3

Factor $2$ out of $0$ .

$\frac{2 (5 \cdot 0) + 2 (0)}{2 + 0 \cdot 5}$

Step 10.1.4

Factor $2$ out of $2 (5 \cdot 0) + 2 (0)$ .

$\frac{2 (5 \cdot 0 + 0)}{2 + 0 \cdot 5}$

Step 10.1.5

Cancel the common factors.

Tap for more steps...

Step 10.1.5.1

Factor $2$ out of $2$ .

$\frac{2 (5 \cdot 0 + 0)}{2 (1) + 0 \cdot 5}$

Step 10.1.5.2

Factor $2$ out of $0 \cdot 5$ .

$\frac{2 (5 \cdot 0 + 0)}{2 (1) + 2 (0 \cdot 5)}$

Step 10.1.5.3

Factor $2$ out of $2 (1) + 2 (0 \cdot 5)$ .

$\frac{2 (5 \cdot 0 + 0)}{2 (1 + 0 \cdot 5)}$

Step 10.1.5.4

Cancel the common factor.

$\frac{2 (5 \cdot 0 + 0)}{2 (1 + 0 \cdot 5)}$

Step 10.1.5.5

Rewrite the expression.

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

Step 10.2

Simplify the numerator.

Tap for more steps...

Step 10.2.1

Multiply $5$ by $0$ .

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

Step 10.2.2

Add $0$ and $0$ .

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

Step 10.3

Simplify the denominator.

Tap for more steps...

Step 10.3.1

Multiply $0$ by $5$ .

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

Step 10.3.2

Add $1$ and $0$ .

$\frac{0}{1}$

Step 10.4

Divide $0$ by $1$ .

$0$