Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit under the radical sign.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify each term.

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Step 6.1

Move $8$ to the left of $v$ .

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

Step 6.2

Multiply $- 1$ by $1$ .

$8 v - 1 - \sqrt{8}$

Step 6.3

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

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Step 6.3.1

Factor $4$ out of $8$ .

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

Step 6.3.2

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

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

Step 6.4

Pull terms out from under the radical.

$8 v - 1 - (2 \sqrt{2})$

Step 6.5

Multiply $2$ by $- 1$ .

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