Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

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

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

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

Step 4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

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

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

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

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

Step 8

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

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

Step 9

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

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

Step 11

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 11.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 11.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 11.4

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\sqrt{\frac{4^{2} - 3 \cdot 4 + 4}{2 \cdot 4^{2} - 4 - 1 \cdot 1}}$

Step 12

Simplify the answer.

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Step 12.1

Raise $4$ to the power of $2$ .

$\sqrt{\frac{16 - 3 \cdot 4 + 4}{2 \cdot 4^{2} - 4 - 1 \cdot 1}}$

Step 12.2

Multiply $- 3$ by $4$ .

$\sqrt{\frac{16 - 12 + 4}{2 \cdot 4^{2} - 4 - 1 \cdot 1}}$

Step 12.3

Subtract $12$ from $16$ .

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

Step 12.4

Add $4$ and $4$ .

$\sqrt{\frac{8}{2 \cdot 4^{2} - 4 - 1 \cdot 1}}$

Step 12.5

Raise $4$ to the power of $2$ .

$\sqrt{\frac{8}{2 \cdot 16 - 4 - 1 \cdot 1}}$

Step 12.6

Multiply $2$ by $16$ .

$\sqrt{\frac{8}{32 - 4 - 1 \cdot 1}}$

Step 12.7

Multiply $- 1$ by $1$ .

$\sqrt{\frac{8}{32 - 4 - 1}}$

Step 12.8

Subtract $4$ from $32$ .

$\sqrt{\frac{8}{28 - 1}}$

Step 12.9

Subtract $1$ from $28$ .

$\sqrt{\frac{8}{27}}$

Step 12.10

Rewrite $\sqrt{\frac{8}{27}}$ as $\frac{\sqrt{8}}{\sqrt{27}}$ .

$\frac{\sqrt{8}}{\sqrt{27}}$

Step 12.11

Simplify the numerator.

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Step 12.11.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 12.11.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)}}{\sqrt{27}}$

Step 12.11.1.2

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

$\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{27}}$

Step 12.11.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{\sqrt{27}}$

Step 12.12

Simplify the denominator.

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Step 12.12.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 12.12.1.1

Factor $9$ out of $27$ .

$\frac{2 \sqrt{2}}{\sqrt{9 (3)}}$

Step 12.12.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{2 \sqrt{2}}{\sqrt{3^{2} \cdot 3}}$

Step 12.12.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{3 \sqrt{3}}$

Step 12.13

Multiply $\frac{2 \sqrt{2}}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 \sqrt{2}}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 12.14

Combine and simplify the denominator.

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Step 12.14.1

Multiply $\frac{2 \sqrt{2}}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 \sqrt{2} \sqrt{3}}{3 \sqrt{3} \sqrt{3}}$

Step 12.14.2

Move $\sqrt{3}$ .

$\frac{2 \sqrt{2} \sqrt{3}}{3 (\sqrt{3} \sqrt{3})}$

Step 12.14.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2 \sqrt{2} \sqrt{3}}{3 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 12.14.4

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2 \sqrt{2} \sqrt{3}}{3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 12.14.5

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

$\frac{2 \sqrt{2} \sqrt{3}}{3 {\sqrt{3}}^{1 + 1}}$

Step 12.14.6

Add $1$ and $1$ .

$\frac{2 \sqrt{2} \sqrt{3}}{3 {\sqrt{3}}^{2}}$

Step 12.14.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 12.14.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{2 \sqrt{2} \sqrt{3}}{3 {(3^{\frac{1}{2}})}^{2}}$

Step 12.14.7.2

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

$\frac{2 \sqrt{2} \sqrt{3}}{3 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 12.14.7.3

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

$\frac{2 \sqrt{2} \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 12.14.7.4

Cancel the common factor of $2$ .

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Step 12.14.7.4.1

Cancel the common factor.

$\frac{2 \sqrt{2} \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 12.14.7.4.2

Rewrite the expression.

$\frac{2 \sqrt{2} \sqrt{3}}{3 \cdot 3^{1}}$

Step 12.14.7.5

Evaluate the exponent.

$\frac{2 \sqrt{2} \sqrt{3}}{3 \cdot 3}$

Step 12.15

Simplify the numerator.

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Step 12.15.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{3 \cdot 2}}{3 \cdot 3}$

Step 12.15.2

Multiply $3$ by $2$ .

$\frac{2 \sqrt{6}}{3 \cdot 3}$

Step 12.16

Multiply $3$ by $3$ .

$\frac{2 \sqrt{6}}{9}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{6}}{9}$

Decimal Form:

$0.54433105 \dots$