Calculus Examples

$lim_{x \to - \infty} \frac{6 x^{2} - x}{\sqrt{9 x^{4} + 7 x^{3}}}$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Factor $x^{3}$ out of $9 x^{4} + 7 x^{3}$ .

Tap for more steps...

Step 1.1.1

Factor $x^{3}$ out of $9 x^{4}$ .

$lim_{x \to - \infty} \frac{6 x^{2} - x}{\sqrt{x^{3} (9 x) + 7 x^{3}}}$

Step 1.1.2

Factor $x^{3}$ out of $7 x^{3}$ .

$lim_{x \to - \infty} \frac{6 x^{2} - x}{\sqrt{x^{3} (9 x) + x^{3} \cdot 7}}$

Step 1.1.3

Factor $x^{3}$ out of $x^{3} (9 x) + x^{3} \cdot 7$ .

$lim_{x \to - \infty} \frac{6 x^{2} - x}{\sqrt{x^{3} (9 x + 7)}}$

Step 1.2

Rewrite $x^{3} (9 x + 7)$ as $x^{2} (x (3^{2} x + 7))$ .

Tap for more steps...

Step 1.2.1

Factor out $x^{2}$ .

$lim_{x \to - \infty} \frac{6 x^{2} - x}{\sqrt{x^{2} x (9 x + 7)}}$

Step 1.2.2

Rewrite $9$ as $3^{2}$ .

$lim_{x \to - \infty} \frac{6 x^{2} - x}{\sqrt{x^{2} x (3^{2} x + 7)}}$

Step 1.2.3

Add parentheses.

$lim_{x \to - \infty} \frac{6 x^{2} - x}{\sqrt{x^{2} (x (3^{2} x + 7))}}$

Step 1.3

Pull terms out from under the radical.

$lim_{x \to - \infty} \frac{6 x^{2} - x}{x \sqrt{x (3^{2} x + 7)}}$

Step 1.4

Raise $3$ to the power of $2$ .

$lim_{x \to - \infty} \frac{6 x^{2} - x}{x \sqrt{x (9 x + 7)}}$

Step 2

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

$lim_{x \to - \infty} \frac{\frac{6 x^{2}}{x^{2}} - \frac{x}{x^{2}}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

Tap for more steps...

Step 3.1.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{6 x^{2}}{x^{2}} - \frac{x}{x^{2}}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3.1.1.2

Divide $6$ by $1$ .

$lim_{x \to - \infty} \frac{6 - \frac{x}{x^{2}}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

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

$lim_{x \to - \infty} \frac{6 - \frac{x^{1}}{x^{2}}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3.1.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to - \infty} \frac{6 - \frac{x \cdot 1}{x^{2}}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3.1.2.3

Cancel the common factors.

Tap for more steps...

Step 3.1.2.3.1

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

$lim_{x \to - \infty} \frac{6 - \frac{x \cdot 1}{x \cdot x}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3.1.2.3.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{6 - \frac{x \cdot 1}{x \cdot x}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3.1.2.3.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{x \sqrt{x (9 x + 7)}}{x^{2}}}$

Step 3.2

Simplify terms.

Tap for more steps...

Step 3.2.1

Simplify terms.

Tap for more steps...

Step 3.2.1.1

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

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{\sqrt{x (9 x + 7)}}{x}}$

Step 3.2.1.2

Apply the distributive property.

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{\sqrt{x (9 x) + x \cdot 7}}{x}}$

Step 3.2.1.3

Reorder.

Tap for more steps...

Step 3.2.1.3.1

Rewrite using the commutative property of multiplication.

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{\sqrt{9 x \cdot x + x \cdot 7}}{x}}$

Step 3.2.1.3.2

Move $7$ to the left of $x$ .

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{\sqrt{9 x \cdot x + 7 \cdot x}}{x}}$

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

Move $x$ .

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{\sqrt{9 (x \cdot x) + 7 \cdot x}}{x}}$

Step 3.2.2.2

Multiply $x$ by $x$ .

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{\sqrt{9 x^{2} + 7 \cdot x}}{x}}$

$lim_{x \to - \infty} \frac{6 - \frac{1}{x}}{\frac{\sqrt{9 x^{2} + 7 x}}{x}}$

Step 3.3

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

$\frac{lim_{x \to - \infty} 6 - \frac{1}{x}}{lim_{x \to - \infty} \frac{\sqrt{9 x^{2} + 7 x}}{x}}$

Step 3.4

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

$\frac{lim_{x \to - \infty} 6 - lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} \frac{\sqrt{9 x^{2} + 7 x}}{x}}$

Step 3.5

Evaluate the limit of $6$ which is constant as $x$ approaches $- \infty$ .

$\frac{6 - lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} \frac{\sqrt{9 x^{2} + 7 x}}{x}}$

Step 4

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

$\frac{6 - 0}{lim_{x \to - \infty} \frac{\sqrt{9 x^{2} + 7 x}}{x}}$

Step 5

Factor $x$ out of $9 x^{2} + 7 x$ .

Tap for more steps...

Step 5.1

Factor $x$ out of $9 x^{2}$ .

$\frac{6 - 0}{lim_{x \to - \infty} \frac{\sqrt{x \cdot 9 x + 7 x}}{x}}$

Step 5.2

Factor $x$ out of $7 x$ .

$\frac{6 - 0}{lim_{x \to - \infty} \frac{\sqrt{x \cdot 9 x + x \cdot 7}}{x}}$

Step 5.3

Factor $x$ out of $x (9 x) + x \cdot 7$ .

$\frac{6 - 0}{lim_{x \to - \infty} \frac{\sqrt{x (9 x + 7)}}{x}}$

Step 6

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

$\frac{6 - 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (9 x + 7)}{x^{2}}}}{\frac{x}{x}}}$

Step 7

Cancel the common factor of $x$ .

$\frac{6 - 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (9 x + 7)}{x^{2}}}}{1}}$

Step 8

Cancel the common factors.

Tap for more steps...

Step 8.1

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

$\frac{6 - 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (9 x + 7)}{x \cdot x}}}{1}}$

Step 8.2

Cancel the common factor.

$\frac{6 - 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{x (9 x + 7)}{x \cdot x}}}{1}}$

Step 8.3

Rewrite the expression.

$\frac{6 - 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{9 x + 7}{x}}}{1}}$

Step 9

Evaluate the limit.

Tap for more steps...

Step 9.1

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

$\frac{6 - 0}{\frac{lim_{x \to - \infty} - \sqrt{\frac{9 x + 7}{x}}}{lim_{x \to - \infty} 1}}$

Step 9.2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 - 0}{\frac{- lim_{x \to - \infty} \sqrt{\frac{9 x + 7}{x}}}{lim_{x \to - \infty} 1}}$

Step 9.3

Move the limit under the radical sign.

$\frac{6 - 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{9 x + 7}{x}}}{lim_{x \to - \infty} 1}}$

Step 10

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

$\frac{6 - 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{9 x}{x} + \frac{7}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}}$

Step 11

Evaluate the limit.

Tap for more steps...

Step 11.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 11.1.1

Cancel the common factor.

$\frac{6 - 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{9 x}{x} + \frac{7}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}}$

Step 11.1.2

Divide $9$ by $1$ .

$\frac{6 - 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{9 + \frac{7}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}}$

Step 11.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 11.2.1

Cancel the common factor.

$\frac{6 - 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{9 + \frac{7}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}}$

Step 11.2.2

Rewrite the expression.

$\frac{6 - 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{9 + \frac{7}{x}}{1}}}{lim_{x \to - \infty} 1}}$

Step 11.3

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

$\frac{6 - 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 + \frac{7}{x}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}$

Step 11.4

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

$\frac{6 - 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 + lim_{x \to - \infty} \frac{7}{x}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}$

Step 11.5

Evaluate the limit of $9$ which is constant as $x$ approaches $- \infty$ .

$\frac{6 - 0}{\frac{- \sqrt{\frac{9 + lim_{x \to - \infty} \frac{7}{x}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}$

Step 11.6

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

$\frac{6 - 0}{\frac{- \sqrt{\frac{9 + 7 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}$

Step 12

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

$\frac{6 - 0}{\frac{- \sqrt{\frac{9 + 7 \cdot 0}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}$

Step 13

Evaluate the limit.

Tap for more steps...

Step 13.1

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

$\frac{6 - 0}{\frac{- \sqrt{\frac{9 + 7 \cdot 0}{1}}}{lim_{x \to - \infty} 1}}$

Step 13.2

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

$\frac{6 - 0}{\frac{- \sqrt{\frac{9 + 7 \cdot 0}{1}}}{1}}$

Step 13.3

Simplify the answer.

Tap for more steps...

Step 13.3.1

Divide $9 + 7 \cdot 0$ by $1$ .

$\frac{6 - 0}{\frac{- \sqrt{9 + 7 \cdot 0}}{1}}$

Step 13.3.2

Divide $- \sqrt{9 + 7 \cdot 0}$ by $1$ .

$\frac{6 - 0}{- \sqrt{9 + 7 \cdot 0}}$

Step 13.3.3

Simplify the numerator.

Tap for more steps...

Step 13.3.3.1

Multiply $- 1$ by $0$ .

$\frac{6 + 0}{- \sqrt{9 + 7 \cdot 0}}$

Step 13.3.3.2

Add $6$ and $0$ .

$\frac{6}{- \sqrt{9 + 7 \cdot 0}}$

Step 13.3.4

Simplify the denominator.

Tap for more steps...

Step 13.3.4.1

Multiply $7$ by $0$ .

$\frac{6}{- \sqrt{9 + 0}}$

Step 13.3.4.2

Add $9$ and $0$ .

$\frac{6}{- \sqrt{9}}$

Step 13.3.4.3

Rewrite $9$ as $3^{2}$ .

$\frac{6}{- \sqrt{3^{2}}}$

Step 13.3.4.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{6}{- 1 \cdot 3}$

Step 13.3.5

Multiply $- 1$ by $3$ .

$\frac{6}{- 3}$

Step 13.3.6

Divide $6$ by $- 3$ .

$- 2$