Calculus Examples

$lim_{x \to 8} (\sqrt{| x |}) - \sqrt{| x - 1 |}$

Step 1

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

$lim_{x \to 8} \sqrt{| x |} - lim_{x \to 8} \sqrt{| x - 1 |}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} | x |} - lim_{x \to 8} \sqrt{| x - 1 |}$

Step 3

Move the limit inside the absolute value signs.

$\sqrt{| lim_{x \to 8} x |} - lim_{x \to 8} \sqrt{| x - 1 |}$

Step 4

Move the limit under the radical sign.

$\sqrt{| lim_{x \to 8} x |} - \sqrt{lim_{x \to 8} | x - 1 |}$

Step 5

Move the limit inside the absolute value signs.

$\sqrt{| lim_{x \to 8} x |} - \sqrt{| lim_{x \to 8} x - 1 |}$

Step 6

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

$\sqrt{| lim_{x \to 8} x |} - \sqrt{| lim_{x \to 8} x - lim_{x \to 8} 1 |}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\sqrt{| lim_{x \to 8} x |} - \sqrt{| lim_{x \to 8} x - 1 \cdot 1 |}$

Step 8

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

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Step 8.1

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

$\sqrt{| 8 |} - \sqrt{| lim_{x \to 8} x - 1 \cdot 1 |}$

Step 8.2

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

$\sqrt{| 8 |} - \sqrt{| 8 - 1 \cdot 1 |}$

Step 9

Simplify each term.

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Step 9.1

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\sqrt{8} - \sqrt{| 8 - 1 \cdot 1 |}$

Step 9.2

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

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Step 9.2.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)} - \sqrt{| 8 - 1 \cdot 1 |}$

Step 9.2.2

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

$\sqrt{2^{2} \cdot 2} - \sqrt{| 8 - 1 \cdot 1 |}$

Step 9.3

Pull terms out from under the radical.

$2 \sqrt{2} - \sqrt{| 8 - 1 \cdot 1 |}$

Step 9.4

Multiply $- 1$ by $1$ .

$2 \sqrt{2} - \sqrt{| 8 - 1 |}$

Step 9.5

Subtract $1$ from $8$ .

$2 \sqrt{2} - \sqrt{| 7 |}$

Step 9.6

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$2 \sqrt{2} - \sqrt{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2} - \sqrt{7}$

Decimal Form:

$0.18267581 \dots$