Calculus Examples

$lim_{x \to 8} \sqrt{2 x + 1} - \sqrt{2 x}$

Step 1

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

$lim_{x \to 8} \sqrt{2 x + 1} - lim_{x \to 8} \sqrt{2 x}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} 2 x + 1} - lim_{x \to 8} \sqrt{2 x}$

Step 3

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

$\sqrt{lim_{x \to 8} 2 x + lim_{x \to 8} 1} - lim_{x \to 8} \sqrt{2 x}$

Step 4

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

$\sqrt{2 lim_{x \to 8} x + lim_{x \to 8} 1} - lim_{x \to 8} \sqrt{2 x}$

Step 5

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

$\sqrt{2 lim_{x \to 8} x + 1} - lim_{x \to 8} \sqrt{2 x}$

Step 6

Move the limit under the radical sign.

$\sqrt{2 lim_{x \to 8} x + 1} - \sqrt{lim_{x \to 8} 2 x}$

Step 7

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

$\sqrt{2 lim_{x \to 8} x + 1} - \sqrt{2 lim_{x \to 8} x}$

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 $x$ .

$\sqrt{2 \cdot 8 + 1} - \sqrt{2 lim_{x \to 8} x}$

Step 8.2

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

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

Step 9

Simplify each term.

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Step 9.1

Multiply $2$ by $8$ .

$\sqrt{16 + 1} - \sqrt{2 \cdot 8}$

Step 9.2

Add $16$ and $1$ .

$\sqrt{17} - \sqrt{2 \cdot 8}$

Step 9.3

Multiply $2$ by $8$ .

$\sqrt{17} - \sqrt{16}$

Step 9.4

Rewrite $16$ as $4^{2}$ .

$\sqrt{17} - \sqrt{4^{2}}$

Step 9.5

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

$\sqrt{17} - 1 \cdot 4$

Step 9.6

Multiply $- 1$ by $4$ .

$\sqrt{17} - 4$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\sqrt{17} - 4$

Decimal Form:

$0.12310562 \dots$