Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Factor $x$ out of $x \cdot 1 + x (3 x)$ .

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

Step 2

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

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

Step 3

Cancel the common factor of $x$ .

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

Step 4

Cancel the common factors.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

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 $\infty$ .

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

Step 5.2

Move the limit under the radical sign.

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

Step 6

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

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

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.

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

Step 7.1.2

Divide $3$ by $1$ .

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

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

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

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

Step 7.4

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

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

Step 8

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

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

Step 9

Evaluate the limit.

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Step 9.1

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 10

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

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

Step 11

Simplify the answer.

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Step 11.1

Divide $0 + 3$ by $1$ .

$\frac{\sqrt{0 + 3}}{4 - 0}$

Step 11.2

Add $0$ and $3$ .

$\frac{\sqrt{3}}{4 - 0}$

Step 11.3

Simplify the denominator.

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Step 11.3.1

Multiply $- 1$ by $0$ .

$\frac{\sqrt{3}}{4 + 0}$

Step 11.3.2

Add $4$ and $0$ .

$\frac{\sqrt{3}}{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{4}$

Decimal Form:

$0.43301270 \dots$