Calculus Examples

$lim_{x \to 8} (1) (\frac{\sqrt{9 x^{2} \cdot (100 x) - 2}}{4 x + 500})$

Step 1

Multiply $\frac{\sqrt{9 x^{2} \cdot (100 x) - 2}}{4 x + 500}$ by $1$ .

$lim_{x \to 8} \frac{\sqrt{9 x^{2} \cdot (100 x) - 2}}{4 x + 500}$

Step 2

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

$\frac{lim_{x \to 8} \sqrt{9 x^{2} \cdot (100 x) - 2}}{lim_{x \to 8} 4 x + 500}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 9 x^{2} \cdot (100 x) - 2}}{lim_{x \to 8} 4 x + 500}$

Step 4

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

$\frac{\sqrt{lim_{x \to 8} 9 x^{2} \cdot 100 x - lim_{x \to 8} 2}}{lim_{x \to 8} 4 x + 500}$

Step 5

Move the term $9 \cdot 100$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{9 \cdot 100 lim_{x \to 8} x^{2} x - lim_{x \to 8} 2}}{lim_{x \to 8} 4 x + 500}$

Step 6

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

$\frac{\sqrt{9 \cdot 100 lim_{x \to 8} x^{2} \cdot lim_{x \to 8} x - lim_{x \to 8} 2}}{lim_{x \to 8} 4 x + 500}$

Step 7

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

$\frac{\sqrt{9 \cdot 100 {(lim_{x \to 8} x)}^{2} \cdot lim_{x \to 8} x - lim_{x \to 8} 2}}{lim_{x \to 8} 4 x + 500}$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{9 \cdot 100 {(lim_{x \to 8} x)}^{2} \cdot lim_{x \to 8} x - 1 \cdot 2}}{lim_{x \to 8} 4 x + 500}$

Step 9

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

$\frac{\sqrt{9 \cdot 100 {(lim_{x \to 8} x)}^{2} \cdot lim_{x \to 8} x - 1 \cdot 2}}{lim_{x \to 8} 4 x + lim_{x \to 8} 500}$

Step 10

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

$\frac{\sqrt{9 \cdot 100 {(lim_{x \to 8} x)}^{2} \cdot lim_{x \to 8} x - 1 \cdot 2}}{4 lim_{x \to 8} x + lim_{x \to 8} 500}$

Step 11

Evaluate the limit of $500$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{9 \cdot 100 {(lim_{x \to 8} x)}^{2} \cdot lim_{x \to 8} x - 1 \cdot 2}}{4 lim_{x \to 8} x + 500}$

Step 12

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

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Step 12.1

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

$\frac{\sqrt{9 \cdot 100 \cdot 8^{2} \cdot lim_{x \to 8} x - 1 \cdot 2}}{4 lim_{x \to 8} x + 500}$

Step 12.2

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

$\frac{\sqrt{9 \cdot 100 \cdot 8^{2} \cdot 8 - 1 \cdot 2}}{4 lim_{x \to 8} x + 500}$

Step 12.3

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

$\frac{\sqrt{9 \cdot 100 \cdot 8^{2} \cdot 8 - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

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Step 13.1.1.1

Move $8$ .

$\frac{\sqrt{9 \cdot 100 \cdot (8 \cdot 8^{2}) - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13.1.1.2

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

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Step 13.1.1.2.1

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

$\frac{\sqrt{9 \cdot 100 \cdot (8^{1} \cdot 8^{2}) - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13.1.1.2.2

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

$\frac{\sqrt{9 \cdot 100 \cdot 8^{1 + 2} - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13.1.1.3

Add $1$ and $2$ .

$\frac{\sqrt{9 \cdot 100 \cdot 8^{3} - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13.1.2

Multiply $9$ by $100$ .

$\frac{\sqrt{900 \cdot 8^{3} - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13.1.3

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

$\frac{\sqrt{900 \cdot 512 - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13.1.4

Multiply $900$ by $512$ .

$\frac{\sqrt{460800 - 1 \cdot 2}}{4 \cdot 8 + 500}$

Step 13.1.5

Multiply $- 1$ by $2$ .

$\frac{\sqrt{460800 - 2}}{4 \cdot 8 + 500}$

Step 13.1.6

Subtract $2$ from $460800$ .

$\frac{\sqrt{460798}}{4 \cdot 8 + 500}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $4$ by $8$ .

$\frac{\sqrt{460798}}{32 + 500}$

Step 13.2.2

Add $32$ and $500$ .

$\frac{\sqrt{460798}}{532}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{460798}}{532}$

Decimal Form:

$1.27597939 \dots$