Calculus Examples

$lim_{x \to 8} 18 - \frac{123 x^{2}}{x^{2} + 40}$

Step 1

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

$lim_{x \to 8} 18 - lim_{x \to 8} \frac{123 x^{2}}{x^{2} + 40}$

Step 2

Evaluate the limit of $18$ which is constant as $x$ approaches $8$ .

$18 - lim_{x \to 8} \frac{123 x^{2}}{x^{2} + 40}$

Step 3

Move the term $123$ outside of the limit because it is constant with respect to $x$ .

$18 - 123 lim_{x \to 8} \frac{x^{2}}{x^{2} + 40}$

Step 4

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

$18 - 123 \frac{lim_{x \to 8} x^{2}}{lim_{x \to 8} x^{2} + 40}$

Step 5

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

$18 - 123 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} x^{2} + 40}$

Step 6

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

$18 - 123 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} x^{2} + lim_{x \to 8} 40}$

Step 7

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

$18 - 123 \frac{{(lim_{x \to 8} x)}^{2}}{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 40}$

Step 8

Evaluate the limit of $40$ which is constant as $x$ approaches $8$ .

$18 - 123 \frac{{(lim_{x \to 8} x)}^{2}}{{(lim_{x \to 8} x)}^{2} + 40}$

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 $x$ .

$18 - 123 \frac{8^{2}}{{(lim_{x \to 8} x)}^{2} + 40}$

Step 9.2

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

$18 - 123 \frac{8^{2}}{8^{2} + 40}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

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

$18 - 123 \frac{64}{8^{2} + 40}$

Step 10.1.2

Simplify the denominator.

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Step 10.1.2.1

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

$18 - 123 \frac{64}{64 + 40}$

Step 10.1.2.2

Add $64$ and $40$ .

$18 - 123 (\frac{64}{104})$

Step 10.1.3

Cancel the common factor of $64$ and $104$ .

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Step 10.1.3.1

Factor $8$ out of $64$ .

$18 - 123 \frac{8 (8)}{104}$

Step 10.1.3.2

Cancel the common factors.

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Step 10.1.3.2.1

Factor $8$ out of $104$ .

$18 - 123 \frac{8 \cdot 8}{8 \cdot 13}$

Step 10.1.3.2.2

Cancel the common factor.

$18 - 123 \frac{8 \cdot 8}{8 \cdot 13}$

Step 10.1.3.2.3

Rewrite the expression.

$18 - 123 (\frac{8}{13})$

Step 10.1.4

Multiply $- 123 (\frac{8}{13})$ .

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Step 10.1.4.1

Combine $- 123$ and $\frac{8}{13}$ .

$18 + \frac{- 123 \cdot 8}{13}$

Step 10.1.4.2

Multiply $- 123$ by $8$ .

$18 + \frac{- 984}{13}$

Step 10.1.5

Move the negative in front of the fraction.

$18 - \frac{984}{13}$

Step 10.2

To write $18$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$18 \cdot \frac{13}{13} - \frac{984}{13}$

Step 10.3

Combine $18$ and $\frac{13}{13}$ .

$\frac{18 \cdot 13}{13} - \frac{984}{13}$

Step 10.4

Combine the numerators over the common denominator.

$\frac{18 \cdot 13 - 984}{13}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Multiply $18$ by $13$ .

$\frac{234 - 984}{13}$

Step 10.5.2

Subtract $984$ from $234$ .

$\frac{- 750}{13}$

Step 10.6

Move the negative in front of the fraction.

$- \frac{750}{13}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{750}{13}$

Decimal Form:

$- 57.69230769 \dots$