Calculus Examples

$lim_{x \to 8} \sqrt{9 x^{2} \cdot (6 x) - 1} - (3 x + 4)$

Step 1

Evaluate the limit of $\sqrt{9 x^{2} \cdot (6 x) - 1} - (3 x + 4)$ by plugging in $8$ for $x$ .

$\sqrt{9 \cdot 8^{2} \cdot (6 \cdot 8) - 1} - (3 \cdot 8 + 4)$

Step 2

Simplify each term.

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Step 2.1

Multiply $8^{2}$ by $8$ by adding the exponents.

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Step 2.1.1

Move $8$ .

$\sqrt{9 \cdot (8 \cdot 8^{2}) \cdot 6 - 1} - (3 \cdot 8 + 4)$

Step 2.1.2

Multiply $8$ by $8^{2}$ .

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Step 2.1.2.1

Raise $8$ to the power of $1$ .

$\sqrt{9 \cdot (8^{1} \cdot 8^{2}) \cdot 6 - 1} - (3 \cdot 8 + 4)$

Step 2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{9 \cdot 8^{1 + 2} \cdot 6 - 1} - (3 \cdot 8 + 4)$

Step 2.1.3

Add $1$ and $2$ .

$\sqrt{9 \cdot 8^{3} \cdot 6 - 1} - (3 \cdot 8 + 4)$

Step 2.2

Raise $8$ to the power of $3$ .

$\sqrt{9 \cdot 512 \cdot 6 - 1} - (3 \cdot 8 + 4)$

Step 2.3

Multiply $9 \cdot 512 \cdot 6$ .

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Step 2.3.1

Multiply $9$ by $512$ .

$\sqrt{4608 \cdot 6 - 1} - (3 \cdot 8 + 4)$

Step 2.3.2

Multiply $4608$ by $6$ .

$\sqrt{27648 - 1} - (3 \cdot 8 + 4)$

Step 2.4

Subtract $1$ from $27648$ .

$\sqrt{27647} - (3 \cdot 8 + 4)$

Step 2.5

Multiply $3$ by $8$ .

$\sqrt{27647} - (24 + 4)$

Step 2.6

Add $24$ and $4$ .

$\sqrt{27647} - 1 \cdot 28$

Step 2.7

Multiply $- 1$ by $28$ .

$\sqrt{27647} - 28$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{27647} - 28$

Decimal Form:

$138.27387046 \dots$