Calculus Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

Add $2 x + 1$ and $0$ .

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Factor using the perfect square rule.

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Step 3.1.1.1

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

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

Step 3.1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 3.1.1.3

Rewrite the polynomial.

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

Step 3.1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 3.1.2

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

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

Step 3.2

Simplify terms.

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Step 3.2.1

Add $x$ and $x$ .

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

Step 3.2.2

Cancel the common factor of $2 x + 1$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

$lim_{x \to \infty} 1$

Step 4

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

$1$