Calculus Examples

$lim_{x \to 12} \frac{\sqrt{x - 3} - 2}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $12$ .

$\frac{lim_{x \to 12} \sqrt{x - 3} - 2}{lim_{x \to 12} x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $12$ .

$\frac{lim_{x \to 12} \sqrt{x - 3} - lim_{x \to 12} 2}{lim_{x \to 12} x}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 12} x - 3} - lim_{x \to 12} 2}{lim_{x \to 12} x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $12$ .

$\frac{\sqrt{lim_{x \to 12} x - lim_{x \to 12} 3} - lim_{x \to 12} 2}{lim_{x \to 12} x}$

Step 5

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

$\frac{\sqrt{lim_{x \to 12} x - 1 \cdot 3} - lim_{x \to 12} 2}{lim_{x \to 12} x}$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $12$ .

$\frac{\sqrt{lim_{x \to 12} x - 1 \cdot 3} - 1 \cdot 2}{lim_{x \to 12} x}$

Step 7

Evaluate the limits by plugging in $12$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $12$ for $x$ .

$\frac{\sqrt{12 - 1 \cdot 3} - 1 \cdot 2}{lim_{x \to 12} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $12$ for $x$ .

$\frac{\sqrt{12 - 1 \cdot 3} - 1 \cdot 2}{12}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $3$ .

$\frac{\sqrt{12 - 3} - 1 \cdot 2}{12}$

Step 8.2

Subtract $3$ from $12$ .

$\frac{\sqrt{9} - 1 \cdot 2}{12}$

Step 8.3

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

$\frac{\sqrt{3^{2}} - 1 \cdot 2}{12}$

Step 8.4

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

$\frac{3 - 1 \cdot 2}{12}$

Step 8.5

Multiply $- 1$ by $2$ .

$\frac{3 - 2}{12}$

Step 8.6

Subtract $2$ from $3$ .

$\frac{1}{12}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{12}$

Decimal Form:

$0.08\overline{3}$