Calculus Examples

$\frac{lim_{x \to 8} x^{5} - 1000 x^{4}}{2 x^{5} + 10000}$

Step 1

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

$\frac{lim_{x \to 8} x^{5} - lim_{x \to 8} 1000 x^{4}}{2 x^{5} + 10000}$

Step 2

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 8} x)}^{5} - lim_{x \to 8} 1000 x^{4}}{2 x^{5} + 10000}$

Step 3

Move the term $1000$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 8} x)}^{5} - 1000 lim_{x \to 8} x^{4}}{2 x^{5} + 10000}$

Step 4

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

$\frac{{(lim_{x \to 8} x)}^{5} - 1000 {(lim_{x \to 8} x)}^{4}}{2 x^{5} + 10000}$

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 $x$ .

$\frac{8^{5} - 1000 {(lim_{x \to 8} x)}^{4}}{2 x^{5} + 10000}$

Step 5.2

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

$\frac{8^{5} - 1000 \cdot 8^{4}}{2 x^{5} + 10000}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $8$ to the power of $5$ .

$\frac{32768 - 1000 \cdot 8^{4}}{2 x^{5} + 10000}$

Step 6.1.2

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

$\frac{32768 - 1000 \cdot 4096}{2 x^{5} + 10000}$

Step 6.1.3

Multiply $- 1000$ by $4096$ .

$\frac{32768 - 4096000}{2 x^{5} + 10000}$

Step 6.1.4

Subtract $4096000$ from $32768$ .

$\frac{- 4063232}{2 x^{5} + 10000}$

Step 6.2

Factor $2$ out of $2 x^{5} + 10000$ .

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Step 6.2.1

Factor $2$ out of $2 x^{5}$ .

$\frac{- 4063232}{2 (x^{5}) + 10000}$

Step 6.2.2

Factor $2$ out of $10000$ .

$\frac{- 4063232}{2 x^{5} + 2 \cdot 5000}$

Step 6.2.3

Factor $2$ out of $2 x^{5} + 2 \cdot 5000$ .

$\frac{- 4063232}{2 (x^{5} + 5000)}$

Step 6.3

Cancel the common factor of $- 4063232$ and $2$ .

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Step 6.3.1

Factor $2$ out of $- 4063232$ .

$\frac{2 \cdot - 2031616}{2 (x^{5} + 5000)}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Cancel the common factor.

$\frac{2 \cdot - 2031616}{2 (x^{5} + 5000)}$

Step 6.3.2.2

Rewrite the expression.

$\frac{- 2031616}{x^{5} + 5000}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{2031616}{x^{5} + 5000}$