Calculus Examples

$lim_{x \to - \infty} \frac{\sqrt{16 x^{4} - 8 x^{2}}}{x^{2} - 2}$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Factor $8 x^{2}$ out of $16 x^{4} - 8 x^{2}$ .

Tap for more steps...

Step 1.1.1

Factor $8 x^{2}$ out of $16 x^{4}$ .

$lim_{x \to - \infty} \frac{\sqrt{8 x^{2} (2 x^{2}) - 8 x^{2}}}{x^{2} - 2}$

Step 1.1.2

Factor $8 x^{2}$ out of $- 8 x^{2}$ .

$lim_{x \to - \infty} \frac{\sqrt{8 x^{2} (2 x^{2}) + 8 x^{2} (- 1)}}{x^{2} - 2}$

Step 1.1.3

Factor $8 x^{2}$ out of $8 x^{2} (2 x^{2}) + 8 x^{2} (- 1)$ .

$lim_{x \to - \infty} \frac{\sqrt{8 x^{2} (2 x^{2} - 1)}}{x^{2} - 2}$

Step 1.2

Rewrite $8 x^{2} (2 x^{2} - 1)$ as ${(2 x)}^{2} \cdot (2 (2 x^{2} - 1))$ .

Tap for more steps...

Step 1.2.1

Factor $4$ out of $8$ .

$lim_{x \to - \infty} \frac{\sqrt{4 (2) x^{2} (2 x^{2} - 1)}}{x^{2} - 2}$

Step 1.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.3

Move $2$ .

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

Step 1.2.4

Rewrite $2^{2} x^{2}$ as ${(2 x)}^{2}$ .

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

Step 1.2.5

Add parentheses.

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

Step 1.3

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Factor $x$ out of $2 x \sqrt{2 (2 x^{2} - 1)}$ .

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

Step 3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.1

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

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Simplify terms.

Tap for more steps...

Step 3.2.1

Simplify each term.

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

Step 3.2.2

Simplify by multiplying through.

Tap for more steps...

Step 3.2.2.1

Apply the distributive property.

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

Step 3.2.2.2

Multiply.

Tap for more steps...

Step 3.2.2.2.1

Multiply $2$ by $2$ .

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

Step 3.2.2.2.2

Multiply $2$ by $- 1$ .

$lim_{x \to - \infty} \frac{\frac{2 \sqrt{4 x^{2} - 2}}{x}}{1 - \frac{2}{x^{2}}}$

Step 3.3

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

$\frac{lim_{x \to - \infty} \frac{2 \sqrt{4 x^{2} - 2}}{x}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 3.4

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

$\frac{2 lim_{x \to - \infty} \frac{\sqrt{4 x^{2} - 2}}{x}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 3.5

Factor $2$ out of $4 x^{2} - 2$ .

Tap for more steps...

Step 3.5.1

Factor $2$ out of $4 x^{2}$ .

$\frac{2 lim_{x \to - \infty} \frac{\sqrt{2 \cdot 2 x^{2} - 2}}{x}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 3.5.2

Factor $2$ out of $- 2$ .

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

Step 3.5.3

Factor $2$ out of $2 (2 x^{2}) + 2 (- 1)$ .

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

Step 4

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

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

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

Cancel the common factor of $x$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Move the limit under the radical sign.

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

Step 5.5

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

$\frac{2 \frac{- \sqrt{2 lim_{x \to - \infty} \frac{2 x^{2} - 1}{x^{2}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 6

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

$\frac{2 \frac{- \sqrt{2 lim_{x \to - \infty} \frac{\frac{2 x^{2}}{x^{2}} - \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 7

Evaluate the limit.

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

Cancel the common factor.

$\frac{2 \frac{- \sqrt{2 lim_{x \to - \infty} \frac{\frac{2 x^{2}}{x^{2}} - \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 7.1.2

Divide $2$ by $1$ .

$\frac{2 \frac{- \sqrt{2 lim_{x \to - \infty} \frac{2 - \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 7.2

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

Tap for more steps...

Step 7.2.1

Cancel the common factor.

$\frac{2 \frac{- \sqrt{2 lim_{x \to - \infty} \frac{2 - \frac{1}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 7.2.2

Rewrite the expression.

$\frac{2 \frac{- \sqrt{2 lim_{x \to - \infty} \frac{2 - \frac{1}{x^{2}}}{1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 7.3

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

$\frac{2 \frac{- \sqrt{2 \frac{lim_{x \to - \infty} 2 - \frac{1}{x^{2}}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 7.4

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

$\frac{2 \frac{- \sqrt{2 \frac{lim_{x \to - \infty} 2 - lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 7.5

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 8

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - 0}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 9

Evaluate the limit.

Tap for more steps...

Step 9.1

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - 0}{1}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 9.2

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - 0}{1}}}{1}}{lim_{x \to - \infty} 1 - \frac{2}{x^{2}}}$

Step 9.3

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - 0}{1}}}{1}}{lim_{x \to - \infty} 1 - lim_{x \to - \infty} \frac{2}{x^{2}}}$

Step 9.4

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - 0}{1}}}{1}}{1 - lim_{x \to - \infty} \frac{2}{x^{2}}}$

Step 9.5

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - 0}{1}}}{1}}{1 - 2 lim_{x \to - \infty} \frac{1}{x^{2}}}$

Step 10

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

$\frac{2 \frac{- \sqrt{2 \frac{2 - 0}{1}}}{1}}{1 - 2 \cdot 0}$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Divide $2 - 0$ by $1$ .

$\frac{2 \frac{- \sqrt{2 (2 - 0)}}{1}}{1 - 2 \cdot 0}$

Step 11.2

Divide $- \sqrt{2 (2 - 0)}$ by $1$ .

$\frac{2 (- \sqrt{2 (2 - 0)})}{1 - 2 \cdot 0}$

Step 11.3

Multiply $- 1$ by $2$ .

$\frac{- 2 \sqrt{2 (2 - 0)}}{1 - 2 \cdot 0}$

Step 11.4

Simplify the denominator.

Tap for more steps...

Step 11.4.1

Multiply $- 2$ by $0$ .

$\frac{- 2 \sqrt{2 (2 - 0)}}{1 + 0}$

Step 11.4.2

Add $1$ and $0$ .

$\frac{- 2 \sqrt{2 (2 - 0)}}{1}$

Step 11.5

Simplify the numerator.

Tap for more steps...

Step 11.5.1

Multiply $- 1$ by $0$ .

$\frac{- 2 \sqrt{2 (2 + 0)}}{1}$

Step 11.5.2

Add $2$ and $0$ .

$\frac{- 2 \sqrt{2 \cdot 2}}{1}$

Step 11.5.3

Multiply $2$ by $2$ .

$\frac{- 2 \sqrt{4}}{1}$

Step 11.5.4

Rewrite $4$ as $2^{2}$ .

$\frac{- 2 \sqrt{2^{2}}}{1}$

Step 11.5.5

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

$\frac{- 2 \cdot 2}{1}$

Step 11.6

Multiply $- 2$ by $2$ .

$\frac{- 4}{1}$

Step 11.7

Divide $- 4$ by $1$ .

$- 4$