Calculus Examples

$lim_{x \to 8} \frac{80000}{\sqrt{7 (1000000000) - 60000}}$

Step 1

Evaluate the limit of $\frac{80000}{\sqrt{7 (1000000000) - 60000}}$ which is constant as $x$ approaches $8$ .

$\frac{80000}{\sqrt{7 (1000000000) - 60000}}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Multiply $7$ by $1000000000$ .

$\frac{80000}{\sqrt{7000000000 - 60000}}$

Step 2.1.2

Subtract $60000$ from $7000000000$ .

$\frac{80000}{\sqrt{6999940000}}$

Step 2.1.3

Rewrite $6999940000$ as $100^{2} \cdot 699994$ .

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Step 2.1.3.1

Factor $10000$ out of $6999940000$ .

$\frac{80000}{\sqrt{10000 (699994)}}$

Step 2.1.3.2

Rewrite $10000$ as $100^{2}$ .

$\frac{80000}{\sqrt{100^{2} \cdot 699994}}$

Step 2.1.4

Pull terms out from under the radical.

$\frac{80000}{100 \sqrt{699994}}$

Step 2.2

Cancel the common factor of $80000$ and $100$ .

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Step 2.2.1

Factor $100$ out of $80000$ .

$\frac{100 \cdot 800}{100 \sqrt{699994}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $100$ out of $100 \sqrt{699994}$ .

$\frac{100 \cdot 800}{100 (\sqrt{699994})}$

Step 2.2.2.2

Cancel the common factor.

$\frac{100 \cdot 800}{100 \sqrt{699994}}$

Step 2.2.2.3

Rewrite the expression.

$\frac{800}{\sqrt{699994}}$

Step 2.3

Multiply $\frac{800}{\sqrt{699994}}$ by $\frac{\sqrt{699994}}{\sqrt{699994}}$ .

$\frac{800}{\sqrt{699994}} \cdot \frac{\sqrt{699994}}{\sqrt{699994}}$

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

Multiply $\frac{800}{\sqrt{699994}}$ by $\frac{\sqrt{699994}}{\sqrt{699994}}$ .

$\frac{800 \sqrt{699994}}{\sqrt{699994} \sqrt{699994}}$

Step 2.4.2

Raise $\sqrt{699994}$ to the power of $1$ .

$\frac{800 \sqrt{699994}}{{\sqrt{699994}}^{1} \sqrt{699994}}$

Step 2.4.3

Raise $\sqrt{699994}$ to the power of $1$ .

$\frac{800 \sqrt{699994}}{{\sqrt{699994}}^{1} {\sqrt{699994}}^{1}}$

Step 2.4.4

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

$\frac{800 \sqrt{699994}}{{\sqrt{699994}}^{1 + 1}}$

Step 2.4.5

Add $1$ and $1$ .

$\frac{800 \sqrt{699994}}{{\sqrt{699994}}^{2}}$

Step 2.4.6

Rewrite ${\sqrt{699994}}^{2}$ as $699994$ .

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Step 2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{699994}$ as $699994^{\frac{1}{2}}$ .

$\frac{800 \sqrt{699994}}{{(699994^{\frac{1}{2}})}^{2}}$

Step 2.4.6.2

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

$\frac{800 \sqrt{699994}}{699994^{\frac{1}{2} \cdot 2}}$

Step 2.4.6.3

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

$\frac{800 \sqrt{699994}}{699994^{\frac{2}{2}}}$

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

$\frac{800 \sqrt{699994}}{699994^{\frac{2}{2}}}$

Step 2.4.6.4.2

Rewrite the expression.

$\frac{800 \sqrt{699994}}{699994^{1}}$

Step 2.4.6.5

Evaluate the exponent.

$\frac{800 \sqrt{699994}}{699994}$

Step 2.5

Cancel the common factor of $800$ and $699994$ .

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Step 2.5.1

Factor $2$ out of $800 \sqrt{699994}$ .

$\frac{2 (400 \sqrt{699994})}{699994}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $699994$ .

$\frac{2 (400 \sqrt{699994})}{2 (349997)}$

Step 2.5.2.2

Cancel the common factor.

$\frac{2 (400 \sqrt{699994})}{2 \cdot 349997}$

Step 2.5.2.3

Rewrite the expression.

$\frac{400 \sqrt{699994}}{349997}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{400 \sqrt{699994}}{349997}$

Decimal Form:

$0.95618698 \dots$