Calculus Examples

$lim_{x \to 8} \frac{\sqrt{16 x^{4} + 64 x^{2} + x^{2}}}{2 x^{2} - 4}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{16 x^{4} + 64 x^{2} + x^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 16 x^{4} + 64 x^{2} + x^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} 16 x^{4} + lim_{x \to 8} 64 x^{2} + lim_{x \to 8} x^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 4

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{16 lim_{x \to 8} x^{4} + lim_{x \to 8} 64 x^{2} + lim_{x \to 8} x^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 64 x^{2} + lim_{x \to 8} x^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 6

Move the term $64$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + 64 lim_{x \to 8} x^{2} + lim_{x \to 8} x^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 7

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

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + 64 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 8

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

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + 64 {(lim_{x \to 8} x)}^{2} + {(lim_{x \to 8} x)}^{2}}}{lim_{x \to 8} 2 x^{2} - 4}$

Step 9

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

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + 64 {(lim_{x \to 8} x)}^{2} + {(lim_{x \to 8} x)}^{2}}}{lim_{x \to 8} 2 x^{2} - lim_{x \to 8} 4}$

Step 10

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

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + 64 {(lim_{x \to 8} x)}^{2} + {(lim_{x \to 8} x)}^{2}}}{2 lim_{x \to 8} x^{2} - lim_{x \to 8} 4}$

Step 11

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

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + 64 {(lim_{x \to 8} x)}^{2} + {(lim_{x \to 8} x)}^{2}}}{2 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 4}$

Step 12

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{16 {(lim_{x \to 8} x)}^{4} + 64 {(lim_{x \to 8} x)}^{2} + {(lim_{x \to 8} x)}^{2}}}{2 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}$

Step 13

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

Tap for more steps...

Step 13.1

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

$\frac{\sqrt{16 \cdot 8^{4} + 64 {(lim_{x \to 8} x)}^{2} + {(lim_{x \to 8} x)}^{2}}}{2 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}$

Step 13.2

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

$\frac{\sqrt{16 \cdot 8^{4} + 64 \cdot 8^{2} + {(lim_{x \to 8} x)}^{2}}}{2 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}$

Step 13.3

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

$\frac{\sqrt{16 \cdot 8^{4} + 64 \cdot 8^{2} + 8^{2}}}{2 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}$

Step 13.4

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

$\frac{\sqrt{16 \cdot 8^{4} + 64 \cdot 8^{2} + 8^{2}}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14

Simplify the answer.

Tap for more steps...

Step 14.1

Simplify the numerator.

Tap for more steps...

Step 14.1.1

Raise $8$ to the power of $4$ .

$\frac{\sqrt{16 \cdot 4096 + 64 \cdot 8^{2} + 8^{2}}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.2

Multiply $16$ by $4096$ .

$\frac{\sqrt{65536 + 64 \cdot 8^{2} + 8^{2}}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.3

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

$\frac{\sqrt{65536 + 64 \cdot 64 + 8^{2}}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.4

Multiply $64$ by $64$ .

$\frac{\sqrt{65536 + 4096 + 8^{2}}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.5

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

$\frac{\sqrt{65536 + 4096 + 64}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.6

Add $65536$ and $4096$ .

$\frac{\sqrt{69632 + 64}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.7

Add $69632$ and $64$ .

$\frac{\sqrt{69696}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.8

Rewrite $69696$ as $264^{2}$ .

$\frac{\sqrt{264^{2}}}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.1.9

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

$\frac{264}{2 \cdot 8^{2} - 1 \cdot 4}$

Step 14.2

Simplify the denominator.

Tap for more steps...

Step 14.2.1

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

$\frac{264}{2 \cdot 64 - 1 \cdot 4}$

Step 14.2.2

Multiply $2$ by $64$ .

$\frac{264}{128 - 1 \cdot 4}$

Step 14.2.3

Multiply $- 1$ by $4$ .

$\frac{264}{128 - 4}$

Step 14.2.4

Subtract $4$ from $128$ .

$\frac{264}{124}$

Step 14.3

Cancel the common factor of $264$ and $124$ .

Tap for more steps...

Step 14.3.1

Factor $4$ out of $264$ .

$\frac{4 (66)}{124}$

Step 14.3.2

Cancel the common factors.

Tap for more steps...

Step 14.3.2.1

Factor $4$ out of $124$ .

$\frac{4 \cdot 66}{4 \cdot 31}$

Step 14.3.2.2

Cancel the common factor.

$\frac{4 \cdot 66}{4 \cdot 31}$

Step 14.3.2.3

Rewrite the expression.

$\frac{66}{31}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{66}{31}$

Decimal Form:

$2.12903225 \dots$