Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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Step 1.1.1

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

$lim_{x \to \infty} \frac{\sqrt{4 x (x) + 4 x}}{4 x + 1}$

Step 1.1.2

Factor $4 x$ out of $4 x$ .

$lim_{x \to \infty} \frac{\sqrt{4 x (x) + 4 x (1)}}{4 x + 1}$

Step 1.1.3

Factor $4 x$ out of $4 x (x) + 4 x (1)$ .

$lim_{x \to \infty} \frac{\sqrt{4 x (x + 1)}}{4 x + 1}$

Step 1.2

Rewrite $4 x (x + 1)$ as $2^{2} (x (x + 1^{2}))$ .

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Step 1.2.1

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

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

Step 1.2.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.3

Add parentheses.

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

Step 1.3

Pull terms out from under the radical.

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

Step 1.4

One to any power is one.

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

Step 2

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

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

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $x$ .

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

Step 3.2

Simplify by multiplying through.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $x$ by $x$ .

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

Step 3.2.2.2

Multiply $x$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 4

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

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

Step 5

Cancel the common factor of $x$ .

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

Step 6

Cancel the common factors.

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

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 $\infty$ .

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

Step 7.2

Move the limit under the radical sign.

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

Step 8

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

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

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.

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

Step 9.1.2

Rewrite the expression.

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

Step 9.2

Cancel the common factor of $x$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 10

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

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

Step 11

Evaluate the limit.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 12

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

$\frac{2 \frac{\sqrt{\frac{1 + 0}{1}}}{1}}{4 + 0}$

Step 13

Simplify the answer.

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Step 13.1

Divide $1 + 0$ by $1$ .

$\frac{2 \frac{\sqrt{1 + 0}}{1}}{4 + 0}$

Step 13.2

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

$\frac{2 \sqrt{1 + 0}}{4 + 0}$

Step 13.3

Cancel the common factor of $2$ and $4 + 0$ .

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Step 13.3.1

Factor $2$ out of $2 \sqrt{1 + 0}$ .

$\frac{2 (\sqrt{1 + 0})}{4 + 0}$

Step 13.3.2

Cancel the common factors.

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Step 13.3.2.1

Factor $2$ out of $4$ .

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

Step 13.3.2.2

Factor $2$ out of $0$ .

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

Step 13.3.2.3

Factor $2$ out of $2 \cdot 2 + 2 \cdot 0$ .

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

Step 13.3.2.4

Cancel the common factor.

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

Step 13.3.2.5

Rewrite the expression.

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

Step 13.4

Simplify the numerator.

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Step 13.4.1

Add $1$ and $0$ .

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

Step 13.4.2

Any root of $1$ is $1$ .

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

Step 13.5

Add $2$ and $0$ .

$\frac{1}{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$