Calculus Examples

$\frac{lim_{x \to 8} 7 x^{9} - 4 x^{5} + 2 x - 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 1

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

$\frac{lim_{x \to 8} 7 x^{9} - lim_{x \to 8} 4 x^{5} + lim_{x \to 8} 2 x - lim_{x \to 8} 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 2

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 lim_{x \to 8} x^{9} - lim_{x \to 8} 4 x^{5} + lim_{x \to 8} 2 x - lim_{x \to 8} 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 3

Move the exponent $9$ from $x^{9}$ outside the limit using the Limits Power Rule.

$\frac{7 {(lim_{x \to 8} x)}^{9} - lim_{x \to 8} 4 x^{5} + lim_{x \to 8} 2 x - lim_{x \to 8} 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 4

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

$\frac{7 {(lim_{x \to 8} x)}^{9} - 4 lim_{x \to 8} x^{5} + lim_{x \to 8} 2 x - lim_{x \to 8} 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 5

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

$\frac{7 {(lim_{x \to 8} x)}^{9} - 4 {(lim_{x \to 8} x)}^{5} + lim_{x \to 8} 2 x - lim_{x \to 8} 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 6

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

$\frac{7 {(lim_{x \to 8} x)}^{9} - 4 {(lim_{x \to 8} x)}^{5} + 2 lim_{x \to 8} x - lim_{x \to 8} 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 7

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

$\frac{7 {(lim_{x \to 8} x)}^{9} - 4 {(lim_{x \to 8} x)}^{5} + 2 lim_{x \to 8} x - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 8

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

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Step 8.1

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

$\frac{7 \cdot 8^{9} - 4 {(lim_{x \to 8} x)}^{5} + 2 lim_{x \to 8} x - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 8.2

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

$\frac{7 \cdot 8^{9} - 4 \cdot 8^{5} + 2 lim_{x \to 8} x - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 8.3

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

$\frac{7 \cdot 8^{9} - 4 \cdot 8^{5} + 2 \cdot 8 - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $8$ to the power of $9$ .

$\frac{7 \cdot 134217728 - 4 \cdot 8^{5} + 2 \cdot 8 - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.2

Multiply $7$ by $134217728$ .

$\frac{939524096 - 4 \cdot 8^{5} + 2 \cdot 8 - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.3

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

$\frac{939524096 - 4 \cdot 32768 + 2 \cdot 8 - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.4

Multiply $- 4$ by $32768$ .

$\frac{939524096 - 131072 + 2 \cdot 8 - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.5

Multiply $2$ by $8$ .

$\frac{939524096 - 131072 + 16 - 1 \cdot 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.6

Multiply $- 1$ by $13$ .

$\frac{939524096 - 131072 + 16 - 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.7

Subtract $131072$ from $939524096$ .

$\frac{939393024 + 16 - 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.8

Add $939393024$ and $16$ .

$\frac{939393040 - 13}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.1.9

Subtract $13$ from $939393040$ .

$\frac{939393027}{- 3 x^{9} + x^{8} - 5 x^{2} + 2 x}$

Step 9.2

Factor $x$ out of $- 3 x^{9} + x^{8} - 5 x^{2} + 2 x$ .

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Step 9.2.1

Factor $x$ out of $- 3 x^{9}$ .

$\frac{939393027}{x (- 3 x^{8}) + x^{8} - 5 x^{2} + 2 x}$

Step 9.2.2

Factor $x$ out of $x^{8}$ .

$\frac{939393027}{x (- 3 x^{8}) + x \cdot x^{7} - 5 x^{2} + 2 x}$

Step 9.2.3

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

$\frac{939393027}{x (- 3 x^{8}) + x \cdot x^{7} + x (- 5 x) + 2 x}$

Step 9.2.4

Factor $x$ out of $2 x$ .

$\frac{939393027}{x (- 3 x^{8}) + x \cdot x^{7} + x (- 5 x) + x \cdot 2}$

Step 9.2.5

Factor $x$ out of $x (- 3 x^{8}) + x \cdot x^{7}$ .

$\frac{939393027}{x (- 3 x^{8} + x^{7}) + x (- 5 x) + x \cdot 2}$

Step 9.2.6

Factor $x$ out of $x (- 3 x^{8} + x^{7}) + x (- 5 x)$ .

$\frac{939393027}{x (- 3 x^{8} + x^{7} - 5 x) + x \cdot 2}$

Step 9.2.7

Factor $x$ out of $x (- 3 x^{8} + x^{7} - 5 x) + x \cdot 2$ .

$\frac{939393027}{x (- 3 x^{8} + x^{7} - 5 x + 2)}$

Step 9.3

Factor $- 1$ out of $- 3 x^{8}$ .

$\frac{939393027}{x (- (3 x^{8}) + x^{7} - 5 x + 2)}$

Step 9.4

Factor $- 1$ out of $x^{7}$ .

$\frac{939393027}{x (- (3 x^{8}) - 1 (- x^{7}) - 5 x + 2)}$

Step 9.5

Factor $- 1$ out of $- (3 x^{8}) - 1 (- x^{7})$ .

$\frac{939393027}{x (- (3 x^{8} - x^{7}) - 5 x + 2)}$

Step 9.6

Factor $- 1$ out of $- 5 x$ .

$\frac{939393027}{x (- (3 x^{8} - x^{7}) - (5 x) + 2)}$

Step 9.7

Factor $- 1$ out of $- (3 x^{8} - x^{7}) - (5 x)$ .

$\frac{939393027}{x (- (3 x^{8} - x^{7} + 5 x) + 2)}$

Step 9.8

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{939393027}{x (- (3 x^{8} - x^{7} + 5 x) - 1 (- 2))}$

Step 9.9

Factor $- 1$ out of $- (3 x^{8} - x^{7} + 5 x) - 1 (- 2)$ .

$\frac{939393027}{x (- (3 x^{8} - x^{7} + 5 x - 2))}$

Step 9.10

Rewrite $- (3 x^{8} - x^{7} + 5 x - 2)$ as $- 1 (3 x^{8} - x^{7} + 5 x - 2)$ .

$\frac{939393027}{x (- 1 (3 x^{8} - x^{7} + 5 x - 2))}$

Step 9.11

Move the negative in front of the fraction.

$- \frac{939393027}{x (3 x^{8} - x^{7} + 5 x - 2)}$