Calculus Examples

$lim_{x \to 8} \frac{\sqrt{12 - x}}{\sqrt{24 - x}} - \frac{2}{4}$

Step 1

Cancel the common factor of $2$ and $4$ .

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Step 1.1

Factor $2$ out of $2$ .

$lim_{x \to 8} \frac{\sqrt{12 - x}}{\sqrt{24 - x}} - \frac{2 (1)}{4}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $4$ .

$lim_{x \to 8} \frac{\sqrt{12 - x}}{\sqrt{24 - x}} - \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to 8} \frac{\sqrt{12 - x}}{\sqrt{24 - x}} - \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.2.3

Rewrite the expression.

$lim_{x \to 8} \frac{\sqrt{12 - x}}{\sqrt{24 - x}} - \frac{1}{2}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} \frac{\sqrt{12 - x}}{\sqrt{24 - x}} - lim_{x \to 8} \frac{1}{2}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \sqrt{12 - x}}{lim_{x \to 8} \sqrt{24 - x}} - lim_{x \to 8} \frac{1}{2}$

Step 4

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 12 - x}}{lim_{x \to 8} \sqrt{24 - x}} - lim_{x \to 8} \frac{1}{2}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{\sqrt{lim_{x \to 8} 12 - lim_{x \to 8} x}}{lim_{x \to 8} \sqrt{24 - x}} - lim_{x \to 8} \frac{1}{2}$

Step 6

Evaluate the limit of $12$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{12 - lim_{x \to 8} x}}{lim_{x \to 8} \sqrt{24 - x}} - lim_{x \to 8} \frac{1}{2}$

Step 7

Move the limit under the radical sign.

$\frac{\sqrt{12 - lim_{x \to 8} x}}{\sqrt{lim_{x \to 8} 24 - x}} - lim_{x \to 8} \frac{1}{2}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{\sqrt{12 - lim_{x \to 8} x}}{\sqrt{lim_{x \to 8} 24 - lim_{x \to 8} x}} - lim_{x \to 8} \frac{1}{2}$

Step 9

Evaluate the limit of $24$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{12 - lim_{x \to 8} x}}{\sqrt{24 - lim_{x \to 8} x}} - lim_{x \to 8} \frac{1}{2}$

Step 10

Evaluate the limit of $\frac{1}{2}$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{12 - lim_{x \to 8} x}}{\sqrt{24 - lim_{x \to 8} x}} - \frac{1}{2}$

Step 11

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{12 - 8}}{\sqrt{24 - lim_{x \to 8} x}} - \frac{1}{2}$

Step 11.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{12 - 8}}{\sqrt{24 - 8}} - \frac{1}{2}$

Step 12

Simplify the answer.

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Step 12.1

Subtract $8$ from $12$ .

$\frac{\sqrt{4}}{\sqrt{24 - 8}} - \frac{1}{2}$

Step 12.2

Simplify each term.

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Step 12.2.1

Combine $\sqrt{4}$ and $\sqrt{24 - 8}$ into a single radical.

$\sqrt{\frac{4}{24 - 8}} - \frac{1}{2}$

Step 12.2.2

Subtract $8$ from $24$ .

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

Step 12.2.3

Cancel the common factor of $4$ and $16$ .

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Step 12.2.3.1

Factor $4$ out of $4$ .

$\sqrt{\frac{4 (1)}{16}} - \frac{1}{2}$

Step 12.2.3.2

Cancel the common factors.

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Step 12.2.3.2.1

Factor $4$ out of $16$ .

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

Step 12.2.3.2.2

Cancel the common factor.

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

Step 12.2.3.2.3

Rewrite the expression.

$\sqrt{\frac{1}{4}} - \frac{1}{2}$

Step 12.2.4

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{\sqrt{1}}{\sqrt{4}} - \frac{1}{2}$

Step 12.2.5

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{4}} - \frac{1}{2}$

Step 12.2.6

Simplify the denominator.

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Step 12.2.6.1

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

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

Step 12.2.6.2

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

$\frac{1}{2} - \frac{1}{2}$

Step 12.3

Combine the numerators over the common denominator.

$\frac{1 - 1}{2}$

Step 12.4

Subtract $1$ from $1$ .

$\frac{0}{2}$

Step 12.5

Divide $0$ by $2$ .

$0$