Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Factor $2$ out of $2$ .

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

Step 1.3

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

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

Step 2

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

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

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $x$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

Move the limit under the radical sign.

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

Step 5.3

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

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

Step 6

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

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

Step 7

Evaluate the limit.

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Step 7.1

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

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Divide $2$ by $1$ .

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

Step 7.2

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

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 8

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

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

Step 9

Evaluate the limit.

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Step 9.1

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

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

Step 9.2

Simplify the answer.

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Step 9.2.1

Divide $2 + 0$ by $1$ .

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

Step 9.2.2

Simplify the numerator.

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Step 9.2.2.1

Multiply $- 1$ by $0$ .

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

Step 9.2.2.2

Add $2$ and $0$ .

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

Step 9.2.3

Simplify the denominator.

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Step 9.2.3.1

Add $2$ and $0$ .

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

Step 9.2.3.2

Multiply $2$ by $2$ .

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

Step 9.2.3.3

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

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

Step 9.2.3.4

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

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

Step 9.2.4

Multiply $- 1$ by $2$ .

$\frac{2}{- 2}$

Step 9.2.5

Divide $2$ by $- 2$ .

$- 1$