Calculus Examples

$lim_{x \to 8} x^{- 8} - 20000 x^{- 4}$

Step 1

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

$lim_{x \to 8} x^{- 8} - lim_{x \to 8} 20000 x^{- 4}$

Step 2

Move the exponent $- 8$ from $x^{- 8}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 8} x)}^{- 8} - lim_{x \to 8} 20000 x^{- 4}$

Step 3

Move the term $20000$ outside of the limit because it is constant with respect to $x$ .

${(lim_{x \to 8} x)}^{- 8} - 20000 lim_{x \to 8} x^{- 4}$

Step 4

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

${(lim_{x \to 8} x)}^{- 8} - 20000 {(lim_{x \to 8} x)}^{- 4}$

Step 5

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

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Step 5.1

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

$8^{- 8} - 20000 {(lim_{x \to 8} x)}^{- 4}$

Step 5.2

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

$8^{- 8} - 20000 \cdot 8^{- 4}$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$\frac{1}{8^{8}} - 20000 \cdot 8^{- 4}$

Step 6.1.2

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

$\frac{1}{16777216} - 20000 \cdot 8^{- 4}$

Step 6.1.3

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

$\frac{1}{16777216} - 20000 \cdot \frac{1}{8^{4}}$

Step 6.1.4

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

$\frac{1}{16777216} - 20000 \cdot \frac{1}{4096}$

Step 6.1.5

Cancel the common factor of $32$ .

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Step 6.1.5.1

Factor $32$ out of $- 20000$ .

$\frac{1}{16777216} + 32 (- 625) \cdot \frac{1}{4096}$

Step 6.1.5.2

Factor $32$ out of $4096$ .

$\frac{1}{16777216} + 32 \cdot - 625 \cdot \frac{1}{32 \cdot 128}$

Step 6.1.5.3

Cancel the common factor.

$\frac{1}{16777216} + 32 \cdot - 625 \cdot \frac{1}{32 \cdot 128}$

Step 6.1.5.4

Rewrite the expression.

$\frac{1}{16777216} - 625 \cdot \frac{1}{128}$

Step 6.1.6

Combine $- 625$ and $\frac{1}{128}$ .

$\frac{1}{16777216} + \frac{- 625}{128}$

Step 6.1.7

Move the negative in front of the fraction.

$\frac{1}{16777216} - \frac{625}{128}$

Step 6.2

To write $- \frac{625}{128}$ as a fraction with a common denominator, multiply by $\frac{131072}{131072}$ .

$\frac{1}{16777216} - \frac{625}{128} \cdot \frac{131072}{131072}$

Step 6.3

Write each expression with a common denominator of $16777216$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.1

Multiply $\frac{625}{128}$ by $\frac{131072}{131072}$ .

$\frac{1}{16777216} - \frac{625 \cdot 131072}{128 \cdot 131072}$

Step 6.3.2

Multiply $128$ by $131072$ .

$\frac{1}{16777216} - \frac{625 \cdot 131072}{16777216}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{1 - 625 \cdot 131072}{16777216}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $- 625$ by $131072$ .

$\frac{1 - 81920000}{16777216}$

Step 6.5.2

Subtract $81920000$ from $1$ .

$\frac{- 81919999}{16777216}$

Step 6.6

Move the negative in front of the fraction.

$- \frac{81919999}{16777216}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{81919999}{16777216}$

Decimal Form:

$- 4.88281244 \dots$