Calculus Examples

$lim_{x \to - \infty} (\sqrt{4 x^{2} + 3 x}) + 2 x$

Step 1

Multiply to rationalize the numerator.

$lim_{x \to - \infty} \frac{((\sqrt{4 x^{2} + 3 x}) + 2 x) (\sqrt{4 x^{2} + 3 x} - 2 x)}{\sqrt{4 x^{2} + 3 x} - 2 x}$

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

$lim_{x \to - \infty} \frac{{\sqrt{4 x^{2} + 3 x}}^{2} + \sqrt{4 x^{2} + 3 x} (- 2 x) + \sqrt{4 x^{2} + 3 x} (2 x) - 4 x^{2}}{\sqrt{4 x^{2} + 3 x} - 2 x}$

Step 2.2

Simplify.

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Step 2.2.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$lim_{x \to - \infty} \frac{3 x + 0}{\sqrt{4 x^{2} + 3 x} - 2 x}$

Step 2.2.2

Add $3 x$ and $0$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Factor $x$ out of $4 x^{2} + 3 x$ .

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Step 3.1.1

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

$lim_{x \to - \infty} \frac{3 x}{\sqrt{x (4 x) + 3 x} - 2 x}$

Step 3.1.2

Factor $x$ out of $3 x$ .

$lim_{x \to - \infty} \frac{3 x}{\sqrt{x (4 x) + x \cdot 3} - 2 x}$

Step 3.1.3

Factor $x$ out of $x (4 x) + x \cdot 3$ .

$lim_{x \to - \infty} \frac{3 x}{\sqrt{x (4 x + 3)} - 2 x}$

Step 3.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to - \infty} \frac{x}{\sqrt{x (4 x + 3)} - 2 x}$

Step 4

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

$3 lim_{x \to - \infty} \frac{\frac{x}{x}}{- \sqrt{\frac{x (4 x + 3)}{x^{2}}} + \frac{- 2 x}{x}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $x$ .

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Step 5.1.1

Cancel the common factor.

$3 lim_{x \to - \infty} \frac{\frac{x}{x}}{- \sqrt{\frac{x (4 x + 3)}{x^{2}}} + \frac{- 2 x}{x}}$

Step 5.1.2

Rewrite the expression.

$3 lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x (4 x + 3)}{x^{2}}} + \frac{- 2 x}{x}}$

Step 5.2

Cancel the common factor of $x$ .

$3 lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x (4 x + 3)}{x^{2}}} - 2}$

Step 6

Cancel the common factors.

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Step 6.1

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

$3 lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x (4 x + 3)}{x \cdot x}} - 2}$

Step 6.2

Cancel the common factor.

$3 lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x (4 x + 3)}{x \cdot x}} - 2}$

Step 6.3

Rewrite the expression.

$3 lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{4 x + 3}{x}} - 2}$

Step 7

Evaluate the limit.

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Step 7.1

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

$3 \frac{lim_{x \to - \infty} 1}{lim_{x \to - \infty} - \sqrt{\frac{4 x + 3}{x}} - 2}$

Step 7.2

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

$3 \frac{1}{lim_{x \to - \infty} - \sqrt{\frac{4 x + 3}{x}} - 2}$

Step 7.3

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

$3 \frac{1}{- lim_{x \to - \infty} \sqrt{\frac{4 x + 3}{x}} - lim_{x \to - \infty} 2}$

Step 7.4

Move the limit under the radical sign.

$3 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{4 x + 3}{x}} - lim_{x \to - \infty} 2}$

Step 8

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

$3 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{\frac{4 x}{x} + \frac{3}{x}}{\frac{x}{x}}} - lim_{x \to - \infty} 2}$

Step 9

Evaluate the limit.

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Step 9.1

Cancel the common factor of $x$ .

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Step 9.1.1

Cancel the common factor.

$3 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{\frac{4 x}{x} + \frac{3}{x}}{\frac{x}{x}}} - lim_{x \to - \infty} 2}$

Step 9.1.2

Divide $4$ by $1$ .

$3 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{4 + \frac{3}{x}}{\frac{x}{x}}} - lim_{x \to - \infty} 2}$

Step 9.2

Cancel the common factor of $x$ .

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Step 9.2.1

Cancel the common factor.

$3 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{4 + \frac{3}{x}}{\frac{x}{x}}} - lim_{x \to - \infty} 2}$

Step 9.2.2

Rewrite the expression.

$3 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{4 + \frac{3}{x}}{1}} - lim_{x \to - \infty} 2}$

Step 9.3

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

$3 \frac{1}{- \sqrt{\frac{lim_{x \to - \infty} 4 + \frac{3}{x}}{lim_{x \to - \infty} 1}} - lim_{x \to - \infty} 2}$

Step 9.4

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

$3 \frac{1}{- \sqrt{\frac{lim_{x \to - \infty} 4 + lim_{x \to - \infty} \frac{3}{x}}{lim_{x \to - \infty} 1}} - lim_{x \to - \infty} 2}$

Step 9.5

Evaluate the limit of $4$ which is constant as $x$ approaches $- \infty$ .

$3 \frac{1}{- \sqrt{\frac{4 + lim_{x \to - \infty} \frac{3}{x}}{lim_{x \to - \infty} 1}} - lim_{x \to - \infty} 2}$

Step 9.6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{1}{- \sqrt{\frac{4 + 3 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}} - lim_{x \to - \infty} 2}$

Step 10

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

$3 \frac{1}{- \sqrt{\frac{4 + 3 \cdot 0}{lim_{x \to - \infty} 1}} - lim_{x \to - \infty} 2}$

Step 11

Evaluate the limit.

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Step 11.1

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

$3 \frac{1}{- \sqrt{\frac{4 + 3 \cdot 0}{1}} - lim_{x \to - \infty} 2}$

Step 11.2

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

$3 \frac{1}{- \sqrt{\frac{4 + 3 \cdot 0}{1}} - 1 \cdot 2}$

Step 11.3

Simplify the answer.

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Step 11.3.1

Divide $4 + 3 \cdot 0$ by $1$ .

$3 \frac{1}{- \sqrt{4 + 3 \cdot 0} - 1 \cdot 2}$

Step 11.3.2

Simplify the denominator.

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Step 11.3.2.1

Multiply $3$ by $0$ .

$3 \frac{1}{- \sqrt{4 + 0} - 1 \cdot 2}$

Step 11.3.2.2

Add $4$ and $0$ .

$3 \frac{1}{- \sqrt{4} - 1 \cdot 2}$

Step 11.3.2.3

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

$3 \frac{1}{- \sqrt{2^{2}} - 1 \cdot 2}$

Step 11.3.2.4

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

$3 \frac{1}{- 1 \cdot 2 - 1 \cdot 2}$

Step 11.3.2.5

Multiply $- 1$ by $2$ .

$3 \frac{1}{- 2 - 1 \cdot 2}$

Step 11.3.2.6

Multiply $- 1$ by $2$ .

$3 \frac{1}{- 2 - 2}$

Step 11.3.2.7

Subtract $2$ from $- 2$ .

$3 (\frac{1}{- 4})$

Step 11.3.3

Move the negative in front of the fraction.

$3 (- \frac{1}{4})$

Step 11.3.4

Multiply $3 (- \frac{1}{4})$ .

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Step 11.3.4.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{1}{4})$

Step 11.3.4.2

Combine $- 3$ and $\frac{1}{4}$ .

$\frac{- 3}{4}$

Step 11.3.5

Move the negative in front of the fraction.

$- \frac{3}{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{4}$

Decimal Form:

$- 0.75$