Calculus Examples

$lim_{x \to \infty} 4 x - \sqrt{16 x^{2} - x}$

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

Step 2.2

Simplify.

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

Step 3

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

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Step 3.1

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

$lim_{x \to \infty} \frac{x}{4 x + \sqrt{x (16 x) - x}}$

Step 3.2

Factor $x$ out of $- x$ .

$lim_{x \to \infty} \frac{x}{4 x + \sqrt{x (16 x) + x \cdot - 1}}$

Step 3.3

Factor $x$ out of $x (16 x) + x \cdot - 1$ .

$lim_{x \to \infty} \frac{x}{4 x + \sqrt{x (16 x - 1)}}$

Step 4

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

$lim_{x \to \infty} \frac{\frac{x}{x}}{\frac{4 x}{x} + \sqrt{\frac{x (16 x - 1)}{x^{2}}}}$

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.

$lim_{x \to \infty} \frac{\frac{x}{x}}{\frac{4 x}{x} + \sqrt{\frac{x (16 x - 1)}{x^{2}}}}$

Step 5.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{1}{\frac{4 x}{x} + \sqrt{\frac{x (16 x - 1)}{x^{2}}}}$

Step 5.2

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{1}{4 + \sqrt{\frac{x (16 x - 1)}{x^{2}}}}$

Step 6

Cancel the common factors.

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Step 6.1

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

$lim_{x \to \infty} \frac{1}{4 + \sqrt{\frac{x (16 x - 1)}{x \cdot x}}}$

Step 6.2

Cancel the common factor.

$lim_{x \to \infty} \frac{1}{4 + \sqrt{\frac{x (16 x - 1)}{x \cdot x}}}$

Step 6.3

Rewrite the expression.

$lim_{x \to \infty} \frac{1}{4 + \sqrt{\frac{16 x - 1}{x}}}$

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 $\infty$ .

$\frac{lim_{x \to \infty} 1}{lim_{x \to \infty} 4 + \sqrt{\frac{16 x - 1}{x}}}$

Step 7.2

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

$\frac{1}{lim_{x \to \infty} 4 + \sqrt{\frac{16 x - 1}{x}}}$

Step 7.3

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

$\frac{1}{lim_{x \to \infty} 4 + lim_{x \to \infty} \sqrt{\frac{16 x - 1}{x}}}$

Step 7.4

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

$\frac{1}{4 + lim_{x \to \infty} \sqrt{\frac{16 x - 1}{x}}}$

Step 7.5

Move the limit under the radical sign.

$\frac{1}{4 + \sqrt{lim_{x \to \infty} \frac{16 x - 1}{x}}}$

Step 8

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

$\frac{1}{4 + \sqrt{lim_{x \to \infty} \frac{\frac{16 x}{x} - \frac{1}{x}}{\frac{x}{x}}}}$

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.

$\frac{1}{4 + \sqrt{lim_{x \to \infty} \frac{\frac{16 x}{x} - \frac{1}{x}}{\frac{x}{x}}}}$

Step 9.1.2

Divide $16$ by $1$ .

$\frac{1}{4 + \sqrt{lim_{x \to \infty} \frac{16 - \frac{1}{x}}{\frac{x}{x}}}}$

Step 9.2

Cancel the common factor of $x$ .

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Step 9.2.1

Cancel the common factor.

$\frac{1}{4 + \sqrt{lim_{x \to \infty} \frac{16 - \frac{1}{x}}{\frac{x}{x}}}}$

Step 9.2.2

Rewrite the expression.

$\frac{1}{4 + \sqrt{lim_{x \to \infty} \frac{16 - \frac{1}{x}}{1}}}$

Step 9.3

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

$\frac{1}{4 + \sqrt{\frac{lim_{x \to \infty} 16 - \frac{1}{x}}{lim_{x \to \infty} 1}}}$

Step 9.4

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

$\frac{1}{4 + \sqrt{\frac{lim_{x \to \infty} 16 - lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}}}$

Step 9.5

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

$\frac{1}{4 + \sqrt{\frac{16 - lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}}}$

Step 10

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

$\frac{1}{4 + \sqrt{\frac{16 - 0}{lim_{x \to \infty} 1}}}$

Step 11

Evaluate the limit.

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Step 11.1

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

$\frac{1}{4 + \sqrt{\frac{16 - 0}{1}}}$

Step 11.2

Simplify the answer.

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Step 11.2.1

Divide $16 - 0$ by $1$ .

$\frac{1}{4 + \sqrt{16 - 0}}$

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

Multiply $- 1$ by $0$ .

$\frac{1}{4 + \sqrt{16 + 0}}$

Step 11.2.2.2

Add $16$ and $0$ .

$\frac{1}{4 + \sqrt{16}}$

Step 11.2.2.3

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

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

Step 11.2.2.4

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

$\frac{1}{4 + 4}$

Step 11.2.2.5

Add $4$ and $4$ .

$\frac{1}{8}$