Calculus Examples

$lim_{x \to - 8} \frac{1}{x} - \frac{8}{x + 33}$

Step 1

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

$lim_{x \to - 8} \frac{1}{x} - lim_{x \to - 8} \frac{8}{x + 33}$

Step 2

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

$\frac{lim_{x \to - 8} 1}{lim_{x \to - 8} x} - lim_{x \to - 8} \frac{8}{x + 33}$

Step 3

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

$\frac{1}{lim_{x \to - 8} x} - lim_{x \to - 8} \frac{8}{x + 33}$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{lim_{x \to - 8} x} - 8 lim_{x \to - 8} \frac{1}{x + 33}$

Step 5

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

$\frac{1}{lim_{x \to - 8} x} - 8 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} x + 33}$

Step 6

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

$\frac{1}{lim_{x \to - 8} x} - 8 \frac{1}{lim_{x \to - 8} x + 33}$

Step 7

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

$\frac{1}{lim_{x \to - 8} x} - 8 \frac{1}{lim_{x \to - 8} x + lim_{x \to - 8} 33}$

Step 8

Evaluate the limit of $33$ which is constant as $x$ approaches $- 8$ .

$\frac{1}{lim_{x \to - 8} x} - 8 \frac{1}{lim_{x \to - 8} x + 33}$

Step 9

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

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Step 9.1

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

$\frac{1}{- 8} - 8 \frac{1}{lim_{x \to - 8} x + 33}$

Step 9.2

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

$\frac{1}{- 8} - 8 \frac{1}{- 8 + 33}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Move the negative in front of the fraction.

$- \frac{1}{8} - 8 \frac{1}{- 8 + 33}$

Step 10.1.2

Add $- 8$ and $33$ .

$- \frac{1}{8} - 8 (\frac{1}{25})$

Step 10.1.3

Combine $- 8$ and $\frac{1}{25}$ .

$- \frac{1}{8} + \frac{- 8}{25}$

Step 10.1.4

Move the negative in front of the fraction.

$- \frac{1}{8} - \frac{8}{25}$

Step 10.2

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$- \frac{1}{8} \cdot \frac{25}{25} - \frac{8}{25}$

Step 10.3

To write $- \frac{8}{25}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{1}{8} \cdot \frac{25}{25} - \frac{8}{25} \cdot \frac{8}{8}$

Step 10.4

Write each expression with a common denominator of $200$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.4.1

Multiply $\frac{1}{8}$ by $\frac{25}{25}$ .

$- \frac{25}{8 \cdot 25} - \frac{8}{25} \cdot \frac{8}{8}$

Step 10.4.2

Multiply $8$ by $25$ .

$- \frac{25}{200} - \frac{8}{25} \cdot \frac{8}{8}$

Step 10.4.3

Multiply $\frac{8}{25}$ by $\frac{8}{8}$ .

$- \frac{25}{200} - \frac{8 \cdot 8}{25 \cdot 8}$

Step 10.4.4

Multiply $25$ by $8$ .

$- \frac{25}{200} - \frac{8 \cdot 8}{200}$

Step 10.5

Combine the numerators over the common denominator.

$\frac{- 25 - 8 \cdot 8}{200}$

Step 10.6

Simplify the numerator.

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Step 10.6.1

Multiply $- 8$ by $8$ .

$\frac{- 25 - 64}{200}$

Step 10.6.2

Subtract $64$ from $- 25$ .

$\frac{- 89}{200}$

Step 10.7

Move the negative in front of the fraction.

$- \frac{89}{200}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{89}{200}$

Decimal Form:

$- 0.445$