Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x^{2}$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $2$ by $1$ .

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

Step 2.2.2

Cancel the common factor of $x$ and $x^{2}$ .

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Step 2.2.2.1

Factor $x$ out of $- x$ .

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

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

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

Step 4

Evaluate the limit.

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Step 4.1

Move the limit under the radical sign.

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $- 2$ by $0$ .

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

Step 7.1.2

Add $1$ and $0$ .

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

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $- 1$ by $0$ .

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

Step 7.2.2

Add $2 + 0$ and $0$ .

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

Step 7.2.3

Add $2$ and $0$ .

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

Step 7.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.4.5

Add $1$ and $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 7.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 7.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.4.6.4

Cancel the common factor of $2$ .

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Step 7.4.6.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.4.6.4.2

Rewrite the expression.

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

Step 7.4.6.5

Evaluate the exponent.

$\frac{\sqrt{2}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$