Calculus Examples

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{\sqrt{16 x^{4} - 5 x^{2}}}$

Step 1

Simplify.

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Step 1.1

Factor $x^{2}$ out of $16 x^{4} - 5 x^{2}$ .

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Step 1.1.1

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

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{\sqrt{x^{2} (16 x^{2}) - 5 x^{2}}}$

Step 1.1.2

Factor $x^{2}$ out of $- 5 x^{2}$ .

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{\sqrt{x^{2} (16 x^{2}) + x^{2} \cdot - 5}}$

Step 1.1.3

Factor $x^{2}$ out of $x^{2} (16 x^{2}) + x^{2} \cdot - 5$ .

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{\sqrt{x^{2} (16 x^{2} - 5)}}$

Step 1.2

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{\sqrt{x^{2} ({(4 x)}^{2} - 5)}}$

Step 1.3

Pull terms out from under the radical.

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{x \sqrt{{(4 x)}^{2} - 5}}$

Step 1.4

Apply the product rule to $4 x$ .

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{x \sqrt{4^{2} x^{2} - 5}}$

Step 1.5

Raise $4$ to the power of $2$ .

$lim_{x \to - \infty} \frac{5 x^{2} + 6 x}{x \sqrt{16 x^{2} - 5}}$

Step 2

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

$lim_{x \to - \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}{\frac{x \sqrt{16 x^{2} - 5}}{x^{2}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{5 x^{2}}{x^{2}} + \frac{6 x}{x^{2}}}{\frac{x \sqrt{16 x^{2} - 5}}{x^{2}}}$

Step 3.1.1.2

Divide $5$ by $1$ .

$lim_{x \to - \infty} \frac{5 + \frac{6 x}{x^{2}}}{\frac{x \sqrt{16 x^{2} - 5}}{x^{2}}}$

Step 3.1.2

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

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Step 3.1.2.1

Factor $x$ out of $6 x$ .

$lim_{x \to - \infty} \frac{5 + \frac{x \cdot 6}{x^{2}}}{\frac{x \sqrt{16 x^{2} - 5}}{x^{2}}}$

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

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

$lim_{x \to - \infty} \frac{5 + \frac{x \cdot 6}{x \cdot x}}{\frac{x \sqrt{16 x^{2} - 5}}{x^{2}}}$

Step 3.1.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{5 + \frac{x \cdot 6}{x \cdot x}}{\frac{x \sqrt{16 x^{2} - 5}}{x^{2}}}$

Step 3.1.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{5 + \frac{6}{x}}{\frac{x \sqrt{16 x^{2} - 5}}{x^{2}}}$

Step 3.2

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

$lim_{x \to - \infty} \frac{5 + \frac{6}{x}}{\frac{\sqrt{16 x^{2} - 5}}{x}}$

Step 3.3

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

$\frac{lim_{x \to - \infty} 5 + \frac{6}{x}}{lim_{x \to - \infty} \frac{\sqrt{16 x^{2} - 5}}{x}}$

Step 3.4

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

$\frac{lim_{x \to - \infty} 5 + lim_{x \to - \infty} \frac{6}{x}}{lim_{x \to - \infty} \frac{\sqrt{16 x^{2} - 5}}{x}}$

Step 3.5

Evaluate the limit of $5$ which is constant as $x$ approaches $- \infty$ .

$\frac{5 + lim_{x \to - \infty} \frac{6}{x}}{lim_{x \to - \infty} \frac{\sqrt{16 x^{2} - 5}}{x}}$

Step 3.6

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 + 6 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} \frac{\sqrt{16 x^{2} - 5}}{x}}$

Step 4

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

$\frac{5 + 6 \cdot 0}{lim_{x \to - \infty} \frac{\sqrt{16 x^{2} - 5}}{x}}$

Step 5

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

$\frac{5 + 6 \cdot 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{16 x^{2}}{x^{2}} + \frac{- 5}{x^{2}}}}{\frac{x}{x}}}$

Step 6

Evaluate the limit.

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Step 6.1

Cancel the common factor of $x$ .

$\frac{5 + 6 \cdot 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{16 x^{2}}{x^{2}} + \frac{- 5}{x^{2}}}}{1}}$

Step 6.2

Simplify each term.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

$\frac{5 + 6 \cdot 0}{lim_{x \to - \infty} \frac{- \sqrt{\frac{16 x^{2}}{x^{2}} + \frac{- 5}{x^{2}}}}{1}}$

Step 6.2.1.2

Divide $16$ by $1$ .

$\frac{5 + 6 \cdot 0}{lim_{x \to - \infty} \frac{- \sqrt{16 + \frac{- 5}{x^{2}}}}{1}}$

Step 6.2.2

Move the negative in front of the fraction.

$\frac{5 + 6 \cdot 0}{lim_{x \to - \infty} \frac{- \sqrt{16 - \frac{5}{x^{2}}}}{1}}$

Step 6.3

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

$\frac{5 + 6 \cdot 0}{\frac{lim_{x \to - \infty} - \sqrt{16 - \frac{5}{x^{2}}}}{lim_{x \to - \infty} 1}}$

Step 6.4

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

$\frac{5 + 6 \cdot 0}{\frac{- lim_{x \to - \infty} \sqrt{16 - \frac{5}{x^{2}}}}{lim_{x \to - \infty} 1}}$

Step 6.5

Move the limit under the radical sign.

$\frac{5 + 6 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} 16 - \frac{5}{x^{2}}}}{lim_{x \to - \infty} 1}}$

Step 6.6

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

$\frac{5 + 6 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} 16 - lim_{x \to - \infty} \frac{5}{x^{2}}}}{lim_{x \to - \infty} 1}}$

Step 6.7

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

$\frac{5 + 6 \cdot 0}{\frac{- \sqrt{16 - lim_{x \to - \infty} \frac{5}{x^{2}}}}{lim_{x \to - \infty} 1}}$

Step 6.8

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

$\frac{5 + 6 \cdot 0}{\frac{- \sqrt{16 - 5 lim_{x \to - \infty} \frac{1}{x^{2}}}}{lim_{x \to - \infty} 1}}$

Step 7

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

$\frac{5 + 6 \cdot 0}{\frac{- \sqrt{16 - 5 \cdot 0}}{lim_{x \to - \infty} 1}}$

Step 8

Evaluate the limit.

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Step 8.1

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

$\frac{5 + 6 \cdot 0}{\frac{- \sqrt{16 - 5 \cdot 0}}{1}}$

Step 8.2

Simplify the answer.

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Step 8.2.1

Divide $- \sqrt{16 - 5 \cdot 0}$ by $1$ .

$\frac{5 + 6 \cdot 0}{- \sqrt{16 - 5 \cdot 0}}$

Step 8.2.2

Simplify the numerator.

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Step 8.2.2.1

Multiply $6$ by $0$ .

$\frac{5 + 0}{- \sqrt{16 - 5 \cdot 0}}$

Step 8.2.2.2

Add $5$ and $0$ .

$\frac{5}{- \sqrt{16 - 5 \cdot 0}}$

Step 8.2.3

Simplify the denominator.

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Step 8.2.3.1

Multiply $- 5$ by $0$ .

$\frac{5}{- \sqrt{16 + 0}}$

Step 8.2.3.2

Add $16$ and $0$ .

$\frac{5}{- \sqrt{16}}$

Step 8.2.3.3

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

$\frac{5}{- \sqrt{4^{2}}}$

Step 8.2.3.4

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

$\frac{5}{- 1 \cdot 4}$

Step 8.2.4

Multiply $- 1$ by $4$ .

$\frac{5}{- 4}$

Step 8.2.5

Move the negative in front of the fraction.

$- \frac{5}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{4}$

Decimal Form:

$- 1.25$