Calculus Examples

$lim_{x \to 8} \frac{x^{2} + 3 x - 2}{- 3 x^{2} + 6}$

Step 1

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

$\frac{lim_{x \to 8} x^{2} + 3 x - 2}{lim_{x \to 8} - 3 x^{2} + 6}$

Step 2

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

$\frac{lim_{x \to 8} x^{2} + lim_{x \to 8} 3 x - lim_{x \to 8} 2}{lim_{x \to 8} - 3 x^{2} + 6}$

Step 3

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

$\frac{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 3 x - lim_{x \to 8} 2}{lim_{x \to 8} - 3 x^{2} + 6}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x - lim_{x \to 8} 2}{lim_{x \to 8} - 3 x^{2} + 6}$

Step 5

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

$\frac{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x - 1 \cdot 2}{lim_{x \to 8} - 3 x^{2} + 6}$

Step 6

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

$\frac{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x - 1 \cdot 2}{- lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 6}$

Step 7

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x - 1 \cdot 2}{- 3 lim_{x \to 8} x^{2} + lim_{x \to 8} 6}$

Step 8

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

$\frac{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x - 1 \cdot 2}{- 3 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 6}$

Step 9

Evaluate the limit of $6$ which is constant as $x$ approaches $8$ .

$\frac{{(lim_{x \to 8} x)}^{2} + 3 lim_{x \to 8} x - 1 \cdot 2}{- 3 {(lim_{x \to 8} x)}^{2} + 6}$

Step 10

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

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Step 10.1

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

$\frac{8^{2} + 3 lim_{x \to 8} x - 1 \cdot 2}{- 3 {(lim_{x \to 8} x)}^{2} + 6}$

Step 10.2

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

$\frac{8^{2} + 3 \cdot 8 - 1 \cdot 2}{- 3 {(lim_{x \to 8} x)}^{2} + 6}$

Step 10.3

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

$\frac{8^{2} + 3 \cdot 8 - 1 \cdot 2}{- 3 \cdot 8^{2} + 6}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{64 + 3 \cdot 8 - 1 \cdot 2}{- 3 \cdot 8^{2} + 6}$

Step 11.1.2

Multiply $3$ by $8$ .

$\frac{64 + 24 - 1 \cdot 2}{- 3 \cdot 8^{2} + 6}$

Step 11.1.3

Multiply $- 1$ by $2$ .

$\frac{64 + 24 - 2}{- 3 \cdot 8^{2} + 6}$

Step 11.1.4

Add $64$ and $24$ .

$\frac{88 - 2}{- 3 \cdot 8^{2} + 6}$

Step 11.1.5

Subtract $2$ from $88$ .

$\frac{86}{- 3 \cdot 8^{2} + 6}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

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

$\frac{86}{- 3 \cdot 64 + 6}$

Step 11.2.2

Multiply $- 3$ by $64$ .

$\frac{86}{- 192 + 6}$

Step 11.2.3

Add $- 192$ and $6$ .

$\frac{86}{- 186}$

Step 11.3

Cancel the common factor of $86$ and $- 186$ .

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Step 11.3.1

Factor $2$ out of $86$ .

$\frac{2 (43)}{- 186}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $2$ out of $- 186$ .

$\frac{2 \cdot 43}{2 \cdot - 93}$

Step 11.3.2.2

Cancel the common factor.

$\frac{2 \cdot 43}{2 \cdot - 93}$

Step 11.3.2.3

Rewrite the expression.

$\frac{43}{- 93}$

Step 11.4

Move the negative in front of the fraction.

$- \frac{43}{93}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{43}{93}$

Decimal Form:

$- 0.46236559 \dots$