Calculus Examples

$lim_{x \to 8} \sqrt{\frac{12 x^{3} - 5 x + 7}{1 + 4 x^{2} + 3 x^{3}}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} \frac{12 x^{3} - 5 x + 7}{1 + 4 x^{2} + 3 x^{3}}}$

Step 2

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

$\sqrt{\frac{lim_{x \to 8} 12 x^{3} - 5 x + 7}{lim_{x \to 8} 1 + 4 x^{2} + 3 x^{3}}}$

Step 3

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

$\sqrt{\frac{lim_{x \to 8} 12 x^{3} - lim_{x \to 8} 5 x + lim_{x \to 8} 7}{lim_{x \to 8} 1 + 4 x^{2} + 3 x^{3}}}$

Step 4

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{12 lim_{x \to 8} x^{3} - lim_{x \to 8} 5 x + lim_{x \to 8} 7}{lim_{x \to 8} 1 + 4 x^{2} + 3 x^{3}}}$

Step 5

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 5 x + lim_{x \to 8} 7}{lim_{x \to 8} 1 + 4 x^{2} + 3 x^{3}}}$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + lim_{x \to 8} 7}{lim_{x \to 8} 1 + 4 x^{2} + 3 x^{3}}}$

Step 7

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + 7}{lim_{x \to 8} 1 + 4 x^{2} + 3 x^{3}}}$

Step 8

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

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + 7}{lim_{x \to 8} 1 + lim_{x \to 8} 4 x^{2} + lim_{x \to 8} 3 x^{3}}}$

Step 9

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

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + 7}{1 + lim_{x \to 8} 4 x^{2} + lim_{x \to 8} 3 x^{3}}}$

Step 10

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + 7}{1 + 4 lim_{x \to 8} x^{2} + lim_{x \to 8} 3 x^{3}}}$

Step 11

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + 7}{1 + 4 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 3 x^{3}}}$

Step 12

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + 7}{1 + 4 {(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x^{3}}}$

Step 13

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\sqrt{\frac{12 {(lim_{x \to 8} x)}^{3} - 5 lim_{x \to 8} x + 7}{1 + 4 {(lim_{x \to 8} x)}^{2} + 3 {(lim_{x \to 8} x)}^{3}}}$

Step 14

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

Tap for more steps...

Step 14.1

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

$\sqrt{\frac{12 \cdot 8^{3} - 5 lim_{x \to 8} x + 7}{1 + 4 {(lim_{x \to 8} x)}^{2} + 3 {(lim_{x \to 8} x)}^{3}}}$

Step 14.2

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

$\sqrt{\frac{12 \cdot 8^{3} - 5 \cdot 8 + 7}{1 + 4 {(lim_{x \to 8} x)}^{2} + 3 {(lim_{x \to 8} x)}^{3}}}$

Step 14.3

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

$\sqrt{\frac{12 \cdot 8^{3} - 5 \cdot 8 + 7}{1 + 4 \cdot 8^{2} + 3 {(lim_{x \to 8} x)}^{3}}}$

Step 14.4

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

$\sqrt{\frac{12 \cdot 8^{3} - 5 \cdot 8 + 7}{1 + 4 \cdot 8^{2} + 3 \cdot 8^{3}}}$

Step 15

Simplify the answer.

Tap for more steps...

Step 15.1

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

$\sqrt{\frac{12 \cdot 512 - 5 \cdot 8 + 7}{1 + 4 \cdot 8^{2} + 3 \cdot 8^{3}}}$

Step 15.2

Multiply $12$ by $512$ .

$\sqrt{\frac{6144 - 5 \cdot 8 + 7}{1 + 4 \cdot 8^{2} + 3 \cdot 8^{3}}}$

Step 15.3

Multiply $- 5$ by $8$ .

$\sqrt{\frac{6144 - 40 + 7}{1 + 4 \cdot 8^{2} + 3 \cdot 8^{3}}}$

Step 15.4

Subtract $40$ from $6144$ .

$\sqrt{\frac{6104 + 7}{1 + 4 \cdot 8^{2} + 3 \cdot 8^{3}}}$

Step 15.5

Add $6104$ and $7$ .

$\sqrt{\frac{6111}{1 + 4 \cdot 8^{2} + 3 \cdot 8^{3}}}$

Step 15.6

Raise $8$ to the power of $2$ .

$\sqrt{\frac{6111}{1 + 4 \cdot 64 + 3 \cdot 8^{3}}}$

Step 15.7

Multiply $4$ by $64$ .

$\sqrt{\frac{6111}{1 + 256 + 3 \cdot 8^{3}}}$

Step 15.8

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

$\sqrt{\frac{6111}{1 + 256 + 3 \cdot 512}}$

Step 15.9

Multiply $3$ by $512$ .

$\sqrt{\frac{6111}{1 + 256 + 1536}}$

Step 15.10

Add $1$ and $256$ .

$\sqrt{\frac{6111}{257 + 1536}}$

Step 15.11

Add $257$ and $1536$ .

$\sqrt{\frac{6111}{1793}}$

Step 15.12

Rewrite $\sqrt{\frac{6111}{1793}}$ as $\frac{\sqrt{6111}}{\sqrt{1793}}$ .

$\frac{\sqrt{6111}}{\sqrt{1793}}$

Step 15.13

Simplify the numerator.

Tap for more steps...

Step 15.13.1

Rewrite $6111$ as $3^{2} \cdot 679$ .

Tap for more steps...

Step 15.13.1.1

Factor $9$ out of $6111$ .

$\frac{\sqrt{9 (679)}}{\sqrt{1793}}$

Step 15.13.1.2

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

$\frac{\sqrt{3^{2} \cdot 679}}{\sqrt{1793}}$

Step 15.13.2

Pull terms out from under the radical.

$\frac{3 \sqrt{679}}{\sqrt{1793}}$

Step 15.14

Multiply $\frac{3 \sqrt{679}}{\sqrt{1793}}$ by $\frac{\sqrt{1793}}{\sqrt{1793}}$ .

$\frac{3 \sqrt{679}}{\sqrt{1793}} \cdot \frac{\sqrt{1793}}{\sqrt{1793}}$

Step 15.15

Combine and simplify the denominator.

Tap for more steps...

Step 15.15.1

Multiply $\frac{3 \sqrt{679}}{\sqrt{1793}}$ by $\frac{\sqrt{1793}}{\sqrt{1793}}$ .

$\frac{3 \sqrt{679} \sqrt{1793}}{\sqrt{1793} \sqrt{1793}}$

Step 15.15.2

Raise $\sqrt{1793}$ to the power of $1$ .

$\frac{3 \sqrt{679} \sqrt{1793}}{{\sqrt{1793}}^{1} \sqrt{1793}}$

Step 15.15.3

Raise $\sqrt{1793}$ to the power of $1$ .

$\frac{3 \sqrt{679} \sqrt{1793}}{{\sqrt{1793}}^{1} {\sqrt{1793}}^{1}}$

Step 15.15.4

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

$\frac{3 \sqrt{679} \sqrt{1793}}{{\sqrt{1793}}^{1 + 1}}$

Step 15.15.5

Add $1$ and $1$ .

$\frac{3 \sqrt{679} \sqrt{1793}}{{\sqrt{1793}}^{2}}$

Step 15.15.6

Rewrite ${\sqrt{1793}}^{2}$ as $1793$ .

Tap for more steps...

Step 15.15.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1793}$ as $1793^{\frac{1}{2}}$ .

$\frac{3 \sqrt{679} \sqrt{1793}}{{(1793^{\frac{1}{2}})}^{2}}$

Step 15.15.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \sqrt{679} \sqrt{1793}}{1793^{\frac{1}{2} \cdot 2}}$

Step 15.15.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 \sqrt{679} \sqrt{1793}}{1793^{\frac{2}{2}}}$

Step 15.15.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.15.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{679} \sqrt{1793}}{1793^{\frac{2}{2}}}$

Step 15.15.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{679} \sqrt{1793}}{1793^{1}}$

Step 15.15.6.5

Evaluate the exponent.

$\frac{3 \sqrt{679} \sqrt{1793}}{1793}$

Step 15.16

Simplify the numerator.

Tap for more steps...

Step 15.16.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{1793 \cdot 679}}{1793}$

Step 15.16.2

Multiply $1793$ by $679$ .

$\frac{3 \sqrt{1217447}}{1793}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{1217447}}{1793}$

Decimal Form:

$1.84614580 \dots$