Calculus Examples

$lim_{x \to 8} 2300 + \frac{125}{x} - \frac{500}{x^{2}}$

Step 1

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

$lim_{x \to 8} 2300 + lim_{x \to 8} \frac{125}{x} - lim_{x \to 8} \frac{500}{x^{2}}$

Step 2

Evaluate the limit of $2300$ which is constant as $x$ approaches $8$ .

$2300 + lim_{x \to 8} \frac{125}{x} - lim_{x \to 8} \frac{500}{x^{2}}$

Step 3

Move the term $125$ outside of the limit because it is constant with respect to $x$ .

$2300 + 125 lim_{x \to 8} \frac{1}{x} - lim_{x \to 8} \frac{500}{x^{2}}$

Step 4

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

$2300 + 125 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{500}{x^{2}}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$2300 + 125 \frac{1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{500}{x^{2}}$

Step 6

Move the term $500$ outside of the limit because it is constant with respect to $x$ .

$2300 + 125 \frac{1}{lim_{x \to 8} x} - 500 lim_{x \to 8} \frac{1}{x^{2}}$

Step 7

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

$2300 + 125 \frac{1}{lim_{x \to 8} x} - 500 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x^{2}}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$2300 + 125 \frac{1}{lim_{x \to 8} x} - 500 \frac{1}{lim_{x \to 8} x^{2}}$

Step 9

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$2300 + 125 \frac{1}{lim_{x \to 8} x} - 500 \frac{1}{{(lim_{x \to 8} x)}^{2}}$

Step 10

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

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Step 10.1

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

$2300 + 125 (\frac{1}{8}) - 500 \frac{1}{{(lim_{x \to 8} x)}^{2}}$

Step 10.2

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

$2300 + 125 (\frac{1}{8}) - 500 \frac{1}{8^{2}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Combine $125$ and $\frac{1}{8}$ .

$2300 + \frac{125}{8} - 500 \frac{1}{8^{2}}$

Step 11.1.2

Raise $8$ to the power of $2$ .

$2300 + \frac{125}{8} - 500 (\frac{1}{64})$

Step 11.1.3

Cancel the common factor of $4$ .

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Step 11.1.3.1

Factor $4$ out of $- 500$ .

$2300 + \frac{125}{8} + 4 (- 125) \frac{1}{64}$

Step 11.1.3.2

Factor $4$ out of $64$ .

$2300 + \frac{125}{8} + 4 \cdot - 125 \frac{1}{4 \cdot 16}$

Step 11.1.3.3

Cancel the common factor.

$2300 + \frac{125}{8} + 4 \cdot - 125 \frac{1}{4 \cdot 16}$

Step 11.1.3.4

Rewrite the expression.

$2300 + \frac{125}{8} - 125 (\frac{1}{16})$

Step 11.1.4

Combine $- 125$ and $\frac{1}{16}$ .

$2300 + \frac{125}{8} + \frac{- 125}{16}$

Step 11.1.5

Move the negative in front of the fraction.

$2300 + \frac{125}{8} - \frac{125}{16}$

Step 11.2

Find the common denominator.

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Step 11.2.1

Write $2300$ as a fraction with denominator $1$ .

$\frac{2300}{1} + \frac{125}{8} - \frac{125}{16}$

Step 11.2.2

Multiply $\frac{2300}{1}$ by $\frac{16}{16}$ .

$\frac{2300}{1} \cdot \frac{16}{16} + \frac{125}{8} - \frac{125}{16}$

Step 11.2.3

Multiply $\frac{2300}{1}$ by $\frac{16}{16}$ .

$\frac{2300 \cdot 16}{16} + \frac{125}{8} - \frac{125}{16}$

Step 11.2.4

Multiply $\frac{125}{8}$ by $\frac{2}{2}$ .

$\frac{2300 \cdot 16}{16} + \frac{125}{8} \cdot \frac{2}{2} - \frac{125}{16}$

Step 11.2.5

Multiply $\frac{125}{8}$ by $\frac{2}{2}$ .

$\frac{2300 \cdot 16}{16} + \frac{125 \cdot 2}{8 \cdot 2} - \frac{125}{16}$

Step 11.2.6

Reorder the factors of $8 \cdot 2$ .

$\frac{2300 \cdot 16}{16} + \frac{125 \cdot 2}{2 \cdot 8} - \frac{125}{16}$

Step 11.2.7

Multiply $2$ by $8$ .

$\frac{2300 \cdot 16}{16} + \frac{125 \cdot 2}{16} - \frac{125}{16}$

Step 11.3

Combine the numerators over the common denominator.

$\frac{2300 \cdot 16 + 125 \cdot 2 - 125}{16}$

Step 11.4

Simplify each term.

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Step 11.4.1

Multiply $2300$ by $16$ .

$\frac{36800 + 125 \cdot 2 - 125}{16}$

Step 11.4.2

Multiply $125$ by $2$ .

$\frac{36800 + 250 - 125}{16}$

Step 11.5

Add $36800$ and $250$ .

$\frac{37050 - 125}{16}$

Step 11.6

Subtract $125$ from $37050$ .

$\frac{36925}{16}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{36925}{16}$

Decimal Form:

$2307.8125$