Calculus Examples

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

Step 1

Factor $x$ out of $16 x^{2} + x$ .

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Step 1.1

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

$lim_{x \to - \infty} \frac{\sqrt{x (16 x) + x}}{x}$

Step 1.2

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

$lim_{x \to - \infty} \frac{\sqrt{x (16 x) + x^{1}}}{x}$

Step 1.3

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

$lim_{x \to - \infty} \frac{\sqrt{x (16 x) + x \cdot 1}}{x}$

Step 1.4

Factor $x$ out of $x (16 x) + x \cdot 1$ .

$lim_{x \to - \infty} \frac{\sqrt{x (16 x + 1)}}{x}$

Step 2

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

$lim_{x \to - \infty} \frac{- \sqrt{\frac{x (16 x + 1)}{x^{2}}}}{\frac{x}{x}}$

Step 3

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{x (16 x + 1)}{x^{2}}}}{1}$

Step 4

Cancel the common factors.

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Step 4.1

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

$lim_{x \to - \infty} \frac{- \sqrt{\frac{x (16 x + 1)}{x \cdot x}}}{1}$

Step 4.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{x (16 x + 1)}{x \cdot x}}}{1}$

Step 4.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{16 x + 1}{x}}}{1}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{lim_{x \to - \infty} - \sqrt{\frac{16 x + 1}{x}}}{lim_{x \to - \infty} 1}$

Step 5.2

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

$\frac{- lim_{x \to - \infty} \sqrt{\frac{16 x + 1}{x}}}{lim_{x \to - \infty} 1}$

Step 5.3

Move the limit under the radical sign.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x + 1}{x}}}{lim_{x \to - \infty} 1}$

Step 6

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

$\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{16 x}{x} + \frac{1}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}$

Step 7

Evaluate the limit.

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Step 7.1

Cancel the common factor of $x$ .

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Step 7.1.1

Cancel the common factor.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{16 x}{x} + \frac{1}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}$

Step 7.1.2

Divide $16$ by $1$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{16 + \frac{1}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}$

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{16 + \frac{1}{x}}{\frac{x}{x}}}}{lim_{x \to - \infty} 1}$

Step 7.2.2

Rewrite the expression.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{16 + \frac{1}{x}}{1}}}{lim_{x \to - \infty} 1}$

Step 7.3

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

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 + \frac{1}{x}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}$

Step 7.4

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

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 + lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}$

Step 7.5

Evaluate the limit of $16$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{\frac{16 + lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}$

Step 8

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

$\frac{- \sqrt{\frac{16 + 0}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 1}$

Step 9

Evaluate the limit.

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Step 9.1

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

$\frac{- \sqrt{\frac{16 + 0}{1}}}{lim_{x \to - \infty} 1}$

Step 9.2

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

$\frac{- \sqrt{\frac{16 + 0}{1}}}{1}$

Step 9.3

Simplify the answer.

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Step 9.3.1

Divide $16 + 0$ by $1$ .

$\frac{- \sqrt{16 + 0}}{1}$

Step 9.3.2

Divide $- \sqrt{16 + 0}$ by $1$ .

$- \sqrt{16 + 0}$

Step 9.3.3

Add $16$ and $0$ .

$- \sqrt{16}$

Step 9.3.4

Rewrite $16$ as $4^{2}$ .

$- \sqrt{4^{2}}$

Step 9.3.5

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

$- 1 \cdot 4$

Step 9.3.6

Multiply $- 1$ by $4$ .

$- 4$