Calculus Examples

$lim_{x \to 3} \frac{\sqrt{12 - x}}{x + 3}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} \sqrt{12 - x}}{lim_{x \to 3} x + 3}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 3} 12 - x}}{lim_{x \to 3} x + 3}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{\sqrt{lim_{x \to 3} 12 - lim_{x \to 3} x}}{lim_{x \to 3} x + 3}$

Step 4

Evaluate the limit of $12$ which is constant as $x$ approaches $3$ .

$\frac{\sqrt{12 - lim_{x \to 3} x}}{lim_{x \to 3} x + 3}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{\sqrt{12 - lim_{x \to 3} x}}{lim_{x \to 3} x + lim_{x \to 3} 3}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{\sqrt{12 - lim_{x \to 3} x}}{lim_{x \to 3} x + 3}$

Step 7

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{\sqrt{12 - 3}}{lim_{x \to 3} x + 3}$

Step 7.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{\sqrt{12 - 3}}{3 + 3}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Subtract $3$ from $12$ .

$\frac{\sqrt{9}}{3 + 3}$

Step 8.1.2

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

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

Step 8.1.3

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

$\frac{3}{3 + 3}$

Step 8.2

Add $3$ and $3$ .

$\frac{3}{6}$

Step 8.3

Cancel the common factor of $3$ and $6$ .

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Step 8.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{6}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $3$ out of $6$ .

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

Step 8.3.2.2

Cancel the common factor.

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

Step 8.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$