Calculus Examples

$lim_{x \to 8} (1) \cdot \frac{1}{115 \sqrt{2 p}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})$

Step 1

Evaluate the limit.

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Step 1.1

Multiply $\frac{1}{115 \sqrt{2 p}}$ by $1$ .

$lim_{x \to 8} \frac{1}{115 \sqrt{2 p}} \cdot (e^{- \frac{1}{2}} {(\frac{x - 512}{115})}^{2})$

Step 1.2

Move the term $\frac{1}{115 \sqrt{2 p}} e^{- \frac{1}{2}}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{115 \sqrt{2 p}} e^{- \frac{1}{2}} lim_{x \to 8} {(\frac{x - 512}{115})}^{2}$

Step 1.3

Move the exponent $2$ from ${(\frac{x - 512}{115})}^{2}$ outside the limit using the Limits Power Rule.

$\frac{1}{115 \sqrt{2 p}} e^{- \frac{1}{2}} {(lim_{x \to 8} \frac{x - 512}{115})}^{2}$

Step 1.4

Move the term $\frac{1}{115}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{115 \sqrt{2 p}} e^{- \frac{1}{2}} {(\frac{1}{115} lim_{x \to 8} x - 512)}^{2}$

Step 1.5

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

$\frac{1}{115 \sqrt{2 p}} e^{- \frac{1}{2}} {(\frac{1}{115} (lim_{x \to 8} x - lim_{x \to 8} 512))}^{2}$

Step 1.6

Evaluate the limit of $512$ which is constant as $x$ approaches $8$ .

$\frac{1}{115 \sqrt{2 p}} e^{- \frac{1}{2}} {(\frac{1}{115} (lim_{x \to 8} x - 1 \cdot 512))}^{2}$

Step 2

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

$\frac{1}{115 \sqrt{2 p}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3

Simplify the answer.

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Step 3.1

Multiply $\frac{1}{115 \sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{1}{115 \sqrt{2 p}} \cdot \frac{\sqrt{2 p}}{\sqrt{2 p}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2

Combine and simplify the denominator.

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Step 3.2.1

Multiply $\frac{1}{115 \sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{\sqrt{2 p}}{115 \sqrt{2 p} \sqrt{2 p}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.2

Move $\sqrt{2 p}$ .

$\frac{\sqrt{2 p}}{115 (\sqrt{2 p} \sqrt{2 p})} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.3

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{\sqrt{2 p}}{115 ({\sqrt{2 p}}^{1} \sqrt{2 p})} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.4

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{\sqrt{2 p}}{115 ({\sqrt{2 p}}^{1} {\sqrt{2 p}}^{1})} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.5

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

$\frac{\sqrt{2 p}}{115 {\sqrt{2 p}}^{1 + 1}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.6

Add $1$ and $1$ .

$\frac{\sqrt{2 p}}{115 {\sqrt{2 p}}^{2}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.7

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

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Step 3.2.7.1

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

$\frac{\sqrt{2 p}}{115 {({(2 p)}^{\frac{1}{2}})}^{2}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.7.2

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

$\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{1}{2} \cdot 2}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.7.3

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

$\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{2}{2}}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.7.4

Cancel the common factor of $2$ .

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Step 3.2.7.4.1

Cancel the common factor.

$\frac{\sqrt{2 p}}{115 {(2 p)}^{\frac{2}{2}}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.7.4.2

Rewrite the expression.

$\frac{\sqrt{2 p}}{115 {(2 p)}^{1}} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.2.7.5

Simplify.

$\frac{\sqrt{2 p}}{115 (2 p)} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.3

Multiply $115$ by $2$ .

$\frac{\sqrt{2 p}}{230 p} e^{- \frac{1}{2}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt{2 p}}{230 p} \cdot \frac{1}{e^{\frac{1}{2}}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.5

Combine.

$\frac{\sqrt{2 p} \cdot 1}{230 p e^{\frac{1}{2}}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.6

Multiply $\sqrt{2 p}$ by $1$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} {(\frac{1}{115} (8 - 1 \cdot 512))}^{2}$

Step 3.7

Multiply $- 1$ by $512$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} {(\frac{1}{115} (8 - 512))}^{2}$

Step 3.8

Subtract $512$ from $8$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} {(\frac{1}{115} \cdot - 504)}^{2}$

Step 3.9

Combine $\frac{1}{115}$ and $- 504$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} {(\frac{- 504}{115})}^{2}$

Step 3.10

Move the negative in front of the fraction.

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} {(- \frac{504}{115})}^{2}$

Step 3.11

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.11.1

Apply the product rule to $- \frac{504}{115}$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} ({(- 1)}^{2} {(\frac{504}{115})}^{2})$

Step 3.11.2

Apply the product rule to $\frac{504}{115}$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} ({(- 1)}^{2} \frac{504^{2}}{115^{2}})$

Step 3.12

Raise $- 1$ to the power of $2$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} (1 \frac{504^{2}}{115^{2}})$

Step 3.13

Multiply $\frac{504^{2}}{115^{2}}$ by $1$ .

$\frac{\sqrt{2 p}}{230 p e^{\frac{1}{2}}} \cdot \frac{504^{2}}{115^{2}}$

Step 3.14

Combine.

$\frac{\sqrt{2 p} \cdot 504^{2}}{230 p e^{\frac{1}{2}} \cdot 115^{2}}$

Step 3.15

Raise $504$ to the power of $2$ .

$\frac{\sqrt{2 p} \cdot 254016}{230 p e^{\frac{1}{2}} \cdot 115^{2}}$

Step 3.16

Simplify the denominator.

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Step 3.16.1

Raise $115$ to the power of $2$ .

$\frac{\sqrt{2 p} \cdot 254016}{230 p e^{\frac{1}{2}} \cdot 13225}$

Step 3.16.2

Multiply $13225$ by $230$ .

$\frac{\sqrt{2 p} \cdot 254016}{3041750 p e^{\frac{1}{2}}}$

Step 3.17

Factor $2$ out of $\sqrt{2 p} \cdot 254016$ .

$\frac{2 (\sqrt{2 p} \cdot 127008)}{3041750 p e^{\frac{1}{2}}}$

Step 3.18

Cancel the common factors.

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Step 3.18.1

Factor $2$ out of $3041750 p e^{\frac{1}{2}}$ .

$\frac{2 (\sqrt{2 p} \cdot 127008)}{2 (1520875 p e^{\frac{1}{2}})}$

Step 3.18.2

Cancel the common factor.

$\frac{2 (\sqrt{2 p} \cdot 127008)}{2 (1520875 p e^{\frac{1}{2}})}$

Step 3.18.3

Rewrite the expression.

$\frac{\sqrt{2 p} \cdot 127008}{1520875 p e^{\frac{1}{2}}}$

Step 3.19

Move $127008$ to the left of $\sqrt{2 p}$ .

$\frac{127008 \sqrt{2 p}}{1520875 p e^{\frac{1}{2}}}$