Calculus Examples

$lim_{x \to 8} \frac{13 - 4 x}{9 + x} + \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 1

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

$lim_{x \to 8} \frac{13 - 4 x}{9 + x} + lim_{x \to 8} \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 2

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

$\frac{lim_{x \to 8} 13 - 4 x}{lim_{x \to 8} 9 + x} + lim_{x \to 8} \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 3

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

$\frac{lim_{x \to 8} 13 - lim_{x \to 8} 4 x}{lim_{x \to 8} 9 + x} + lim_{x \to 8} \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 4

Evaluate the limit of $13$ which is constant as $x$ approaches $8$ .

$\frac{13 - lim_{x \to 8} 4 x}{lim_{x \to 8} 9 + x} + lim_{x \to 8} \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{13 - 4 lim_{x \to 8} x}{lim_{x \to 8} 9 + x} + lim_{x \to 8} \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 6

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

$\frac{13 - 4 lim_{x \to 8} x}{lim_{x \to 8} 9 + lim_{x \to 8} x} + lim_{x \to 8} \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 7

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + lim_{x \to 8} \frac{6 x^{2} + 10}{{(2 x - 5)}^{2}}$

Step 8

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

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{lim_{x \to 8} 6 x^{2} + 10}{lim_{x \to 8} {(2 x - 5)}^{2}}$

Step 9

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

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{lim_{x \to 8} 6 x^{2} + lim_{x \to 8} 10}{lim_{x \to 8} {(2 x - 5)}^{2}}$

Step 10

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{6 lim_{x \to 8} x^{2} + lim_{x \to 8} 10}{lim_{x \to 8} {(2 x - 5)}^{2}}$

Step 11

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

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{6 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 10}{lim_{x \to 8} {(2 x - 5)}^{2}}$

Step 12

Evaluate the limit of $10$ which is constant as $x$ approaches $8$ .

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{6 {(lim_{x \to 8} x)}^{2} + 10}{lim_{x \to 8} {(2 x - 5)}^{2}}$

Step 13

Move the exponent $2$ from ${(2 x - 5)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{6 {(lim_{x \to 8} x)}^{2} + 10}{{(lim_{x \to 8} 2 x - 5)}^{2}}$

Step 14

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

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{6 {(lim_{x \to 8} x)}^{2} + 10}{{(lim_{x \to 8} 2 x - lim_{x \to 8} 5)}^{2}}$

Step 15

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{6 {(lim_{x \to 8} x)}^{2} + 10}{{(2 lim_{x \to 8} x - lim_{x \to 8} 5)}^{2}}$

Step 16

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{13 - 4 lim_{x \to 8} x}{9 + lim_{x \to 8} x} + \frac{6 {(lim_{x \to 8} x)}^{2} + 10}{{(2 lim_{x \to 8} x - 1 \cdot 5)}^{2}}$

Step 17

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

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Step 17.1

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

$\frac{13 - 4 \cdot 8}{9 + lim_{x \to 8} x} + \frac{6 {(lim_{x \to 8} x)}^{2} + 10}{{(2 lim_{x \to 8} x - 1 \cdot 5)}^{2}}$

Step 17.2

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

$\frac{13 - 4 \cdot 8}{9 + 8} + \frac{6 {(lim_{x \to 8} x)}^{2} + 10}{{(2 lim_{x \to 8} x - 1 \cdot 5)}^{2}}$

Step 17.3

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

$\frac{13 - 4 \cdot 8}{9 + 8} + \frac{6 \cdot 8^{2} + 10}{{(2 lim_{x \to 8} x - 1 \cdot 5)}^{2}}$

Step 17.4

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

$\frac{13 - 4 \cdot 8}{9 + 8} + \frac{6 \cdot 8^{2} + 10}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18

Simplify the answer.

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Step 18.1

Multiply $- 4$ by $8$ .

$\frac{13 - 32}{9 + 8} + \frac{6 \cdot 8^{2} + 10}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18.2

Subtract $32$ from $13$ .

$\frac{- 19}{9 + 8} + \frac{6 \cdot 8^{2} + 10}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18.3

Simplify each term.

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Step 18.3.1

Add $9$ and $8$ .

$\frac{- 19}{17} + \frac{6 \cdot 8^{2} + 10}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18.3.2

Move the negative in front of the fraction.

$- \frac{19}{17} + \frac{6 \cdot 8^{2} + 10}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18.3.3

Simplify the numerator.

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Step 18.3.3.1

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

$- \frac{19}{17} + \frac{6 \cdot 64 + 10}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18.3.3.2

Multiply $6$ by $64$ .

$- \frac{19}{17} + \frac{384 + 10}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18.3.3.3

Add $384$ and $10$ .

$- \frac{19}{17} + \frac{394}{{(2 \cdot 8 - 1 \cdot 5)}^{2}}$

Step 18.3.4

Simplify the denominator.

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Step 18.3.4.1

Multiply $2$ by $8$ .

$- \frac{19}{17} + \frac{394}{{(16 - 1 \cdot 5)}^{2}}$

Step 18.3.4.2

Multiply $- 1$ by $5$ .

$- \frac{19}{17} + \frac{394}{{(16 - 5)}^{2}}$

Step 18.3.4.3

Subtract $5$ from $16$ .

$- \frac{19}{17} + \frac{394}{11^{2}}$

Step 18.3.4.4

Raise $11$ to the power of $2$ .

$- \frac{19}{17} + \frac{394}{121}$

Step 18.4

To write $- \frac{19}{17}$ as a fraction with a common denominator, multiply by $\frac{121}{121}$ .

$- \frac{19}{17} \cdot \frac{121}{121} + \frac{394}{121}$

Step 18.5

To write $\frac{394}{121}$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$- \frac{19}{17} \cdot \frac{121}{121} + \frac{394}{121} \cdot \frac{17}{17}$

Step 18.6

Write each expression with a common denominator of $2057$ , by multiplying each by an appropriate factor of $1$ .

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Step 18.6.1

Multiply $\frac{19}{17}$ by $\frac{121}{121}$ .

$- \frac{19 \cdot 121}{17 \cdot 121} + \frac{394}{121} \cdot \frac{17}{17}$

Step 18.6.2

Multiply $17$ by $121$ .

$- \frac{19 \cdot 121}{2057} + \frac{394}{121} \cdot \frac{17}{17}$

Step 18.6.3

Multiply $\frac{394}{121}$ by $\frac{17}{17}$ .

$- \frac{19 \cdot 121}{2057} + \frac{394 \cdot 17}{121 \cdot 17}$

Step 18.6.4

Multiply $121$ by $17$ .

$- \frac{19 \cdot 121}{2057} + \frac{394 \cdot 17}{2057}$

Step 18.7

Combine the numerators over the common denominator.

$\frac{- 19 \cdot 121 + 394 \cdot 17}{2057}$

Step 18.8

Simplify the numerator.

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Step 18.8.1

Multiply $- 19$ by $121$ .

$\frac{- 2299 + 394 \cdot 17}{2057}$

Step 18.8.2

Multiply $394$ by $17$ .

$\frac{- 2299 + 6698}{2057}$

Step 18.8.3

Add $- 2299$ and $6698$ .

$\frac{4399}{2057}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{4399}{2057}$

Decimal Form:

$2.13855128 \dots$