Calculus Examples

$lim_{x \to \frac{3}{8}} {(15 x)}^{- 4} - 20 x^{8}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{3}{8}$ .

$lim_{x \to \frac{3}{8}} {(15 x)}^{- 4} - lim_{x \to \frac{3}{8}} 20 x^{8}$

Step 2

Move the exponent $- 4$ from ${(15 x)}^{- 4}$ outside the limit using the Limits Power Rule.

${(lim_{x \to \frac{3}{8}} 15 x)}^{- 4} - lim_{x \to \frac{3}{8}} 20 x^{8}$

Step 3

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

${(15 lim_{x \to \frac{3}{8}} x)}^{- 4} - lim_{x \to \frac{3}{8}} 20 x^{8}$

Step 4

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

${(15 lim_{x \to \frac{3}{8}} x)}^{- 4} - 20 lim_{x \to \frac{3}{8}} x^{8}$

Step 5

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

${(15 lim_{x \to \frac{3}{8}} x)}^{- 4} - 20 {(lim_{x \to \frac{3}{8}} x)}^{8}$

Step 6

Evaluate the limits by plugging in $\frac{3}{8}$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $\frac{3}{8}$ for $x$ .

${(15 (\frac{3}{8}))}^{- 4} - 20 {(lim_{x \to \frac{3}{8}} x)}^{8}$

Step 6.2

Evaluate the limit of $x$ by plugging in $\frac{3}{8}$ for $x$ .

${(15 (\frac{3}{8}))}^{- 4} - 20 {(\frac{3}{8})}^{8}$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $15 (\frac{3}{8})$ .

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Step 7.1.1.1

Combine $15$ and $\frac{3}{8}$ .

${(\frac{15 \cdot 3}{8})}^{- 4} - 20 {(\frac{3}{8})}^{8}$

Step 7.1.1.2

Multiply $15$ by $3$ .

${(\frac{45}{8})}^{- 4} - 20 {(\frac{3}{8})}^{8}$

Step 7.1.2

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{8}{45})}^{4} - 20 {(\frac{3}{8})}^{8}$

Step 7.1.3

Apply the product rule to $\frac{8}{45}$ .

$\frac{8^{4}}{45^{4}} - 20 {(\frac{3}{8})}^{8}$

Step 7.1.4

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

$\frac{4096}{45^{4}} - 20 {(\frac{3}{8})}^{8}$

Step 7.1.5

Raise $45$ to the power of $4$ .

$\frac{4096}{4100625} - 20 {(\frac{3}{8})}^{8}$

Step 7.1.6

Apply the product rule to $\frac{3}{8}$ .

$\frac{4096}{4100625} - 20 \frac{3^{8}}{8^{8}}$

Step 7.1.7

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

$\frac{4096}{4100625} - 20 \frac{6561}{8^{8}}$

Step 7.1.8

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

$\frac{4096}{4100625} - 20 (\frac{6561}{16777216})$

Step 7.1.9

Cancel the common factor of $4$ .

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Step 7.1.9.1

Factor $4$ out of $- 20$ .

$\frac{4096}{4100625} + 4 (- 5) \frac{6561}{16777216}$

Step 7.1.9.2

Factor $4$ out of $16777216$ .

$\frac{4096}{4100625} + 4 \cdot - 5 \frac{6561}{4 \cdot 4194304}$

Step 7.1.9.3

Cancel the common factor.

$\frac{4096}{4100625} + 4 \cdot - 5 \frac{6561}{4 \cdot 4194304}$

Step 7.1.9.4

Rewrite the expression.

$\frac{4096}{4100625} - 5 (\frac{6561}{4194304})$

Step 7.1.10

Combine $- 5$ and $\frac{6561}{4194304}$ .

$\frac{4096}{4100625} + \frac{- 5 \cdot 6561}{4194304}$

Step 7.1.11

Multiply $- 5$ by $6561$ .

$\frac{4096}{4100625} + \frac{- 32805}{4194304}$

Step 7.1.12

Move the negative in front of the fraction.

$\frac{4096}{4100625} - \frac{32805}{4194304}$

Step 7.2

To write $\frac{4096}{4100625}$ as a fraction with a common denominator, multiply by $\frac{4194304}{4194304}$ .

$\frac{4096}{4100625} \cdot \frac{4194304}{4194304} - \frac{32805}{4194304}$

Step 7.3

To write $- \frac{32805}{4194304}$ as a fraction with a common denominator, multiply by $\frac{4100625}{4100625}$ .

$\frac{4096}{4100625} \cdot \frac{4194304}{4194304} - \frac{32805}{4194304} \cdot \frac{4100625}{4100625}$

Step 7.4

Write each expression with a common denominator of $17199267840000$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.4.1

Multiply $\frac{4096}{4100625}$ by $\frac{4194304}{4194304}$ .

$\frac{4096 \cdot 4194304}{4100625 \cdot 4194304} - \frac{32805}{4194304} \cdot \frac{4100625}{4100625}$

Step 7.4.2

Multiply $4100625$ by $4194304$ .

$\frac{4096 \cdot 4194304}{17199267840000} - \frac{32805}{4194304} \cdot \frac{4100625}{4100625}$

Step 7.4.3

Multiply $\frac{32805}{4194304}$ by $\frac{4100625}{4100625}$ .

$\frac{4096 \cdot 4194304}{17199267840000} - \frac{32805 \cdot 4100625}{4194304 \cdot 4100625}$

Step 7.4.4

Multiply $4194304$ by $4100625$ .

$\frac{4096 \cdot 4194304}{17199267840000} - \frac{32805 \cdot 4100625}{17199267840000}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{4096 \cdot 4194304 - 32805 \cdot 4100625}{17199267840000}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Multiply $4096$ by $4194304$ .

$\frac{17179869184 - 32805 \cdot 4100625}{17199267840000}$

Step 7.6.2

Multiply $- 32805$ by $4100625$ .

$\frac{17179869184 - 134521003125}{17199267840000}$

Step 7.6.3

Subtract $134521003125$ from $17179869184$ .

$\frac{- 117341133941}{17199267840000}$

Step 7.7

Move the negative in front of the fraction.

$- \frac{117341133941}{17199267840000}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{117341133941}{17199267840000}$

Decimal Form:

$- 0.00682244 \dots$