Calculus Examples

$lim_{x \to 8} \frac{5 x^{2} + 7 x - 9}{- 6 x^{2} + 2}$

Step 1

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

$\frac{lim_{x \to 8} 5 x^{2} + 7 x - 9}{lim_{x \to 8} - 6 x^{2} + 2}$

Step 2

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

$\frac{lim_{x \to 8} 5 x^{2} + lim_{x \to 8} 7 x - lim_{x \to 8} 9}{lim_{x \to 8} - 6 x^{2} + 2}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 8} x^{2} + lim_{x \to 8} 7 x - lim_{x \to 8} 9}{lim_{x \to 8} - 6 x^{2} + 2}$

Step 4

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

$\frac{5 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 7 x - lim_{x \to 8} 9}{lim_{x \to 8} - 6 x^{2} + 2}$

Step 5

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

$\frac{5 {(lim_{x \to 8} x)}^{2} + 7 lim_{x \to 8} x - lim_{x \to 8} 9}{lim_{x \to 8} - 6 x^{2} + 2}$

Step 6

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

$\frac{5 {(lim_{x \to 8} x)}^{2} + 7 lim_{x \to 8} x - 1 \cdot 9}{lim_{x \to 8} - 6 x^{2} + 2}$

Step 7

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

$\frac{5 {(lim_{x \to 8} x)}^{2} + 7 lim_{x \to 8} x - 1 \cdot 9}{- lim_{x \to 8} 6 x^{2} + lim_{x \to 8} 2}$

Step 8

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

$\frac{5 {(lim_{x \to 8} x)}^{2} + 7 lim_{x \to 8} x - 1 \cdot 9}{- 6 lim_{x \to 8} x^{2} + lim_{x \to 8} 2}$

Step 9

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

$\frac{5 {(lim_{x \to 8} x)}^{2} + 7 lim_{x \to 8} x - 1 \cdot 9}{- 6 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 2}$

Step 10

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{5 {(lim_{x \to 8} x)}^{2} + 7 lim_{x \to 8} x - 1 \cdot 9}{- 6 {(lim_{x \to 8} x)}^{2} + 2}$

Step 11

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

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Step 11.1

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

$\frac{5 \cdot 8^{2} + 7 lim_{x \to 8} x - 1 \cdot 9}{- 6 {(lim_{x \to 8} x)}^{2} + 2}$

Step 11.2

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

$\frac{5 \cdot 8^{2} + 7 \cdot 8 - 1 \cdot 9}{- 6 {(lim_{x \to 8} x)}^{2} + 2}$

Step 11.3

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

$\frac{5 \cdot 8^{2} + 7 \cdot 8 - 1 \cdot 9}{- 6 \cdot 8^{2} + 2}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

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

$\frac{5 \cdot 64 + 7 \cdot 8 - 1 \cdot 9}{- 6 \cdot 8^{2} + 2}$

Step 12.1.2

Multiply $5$ by $64$ .

$\frac{320 + 7 \cdot 8 - 1 \cdot 9}{- 6 \cdot 8^{2} + 2}$

Step 12.1.3

Multiply $7$ by $8$ .

$\frac{320 + 56 - 1 \cdot 9}{- 6 \cdot 8^{2} + 2}$

Step 12.1.4

Multiply $- 1$ by $9$ .

$\frac{320 + 56 - 9}{- 6 \cdot 8^{2} + 2}$

Step 12.1.5

Add $320$ and $56$ .

$\frac{376 - 9}{- 6 \cdot 8^{2} + 2}$

Step 12.1.6

Subtract $9$ from $376$ .

$\frac{367}{- 6 \cdot 8^{2} + 2}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

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

$\frac{367}{- 6 \cdot 64 + 2}$

Step 12.2.2

Multiply $- 6$ by $64$ .

$\frac{367}{- 384 + 2}$

Step 12.2.3

Add $- 384$ and $2$ .

$\frac{367}{- 382}$

Step 12.3

Move the negative in front of the fraction.

$- \frac{367}{382}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{367}{382}$

Decimal Form:

$- 0.96073298 \dots$