Calculus Examples

$lim_{x \to 3} \frac{2 x^{2} - 5 x - 12}{6 - x - x^{2}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 2 x^{2} - 5 x - 12}{lim_{x \to 3} 6 - x - x^{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 2 x^{2} - lim_{x \to 3} 5 x - lim_{x \to 3} 12}{lim_{x \to 3} 6 - x - x^{2}}$

Step 3

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

$\frac{2 lim_{x \to 3} x^{2} - lim_{x \to 3} 5 x - lim_{x \to 3} 12}{lim_{x \to 3} 6 - x - x^{2}}$

Step 4

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

$\frac{2 {(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 5 x - lim_{x \to 3} 12}{lim_{x \to 3} 6 - x - x^{2}}$

Step 5

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

$\frac{2 {(lim_{x \to 3} x)}^{2} - 5 lim_{x \to 3} x - lim_{x \to 3} 12}{lim_{x \to 3} 6 - x - x^{2}}$

Step 6

Evaluate the limit of $12$ which is constant as $x$ approaches $3$ .

$\frac{2 {(lim_{x \to 3} x)}^{2} - 5 lim_{x \to 3} x - 1 \cdot 12}{lim_{x \to 3} 6 - x - x^{2}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{2 {(lim_{x \to 3} x)}^{2} - 5 lim_{x \to 3} x - 1 \cdot 12}{lim_{x \to 3} 6 - lim_{x \to 3} x - lim_{x \to 3} x^{2}}$

Step 8

Evaluate the limit of $6$ which is constant as $x$ approaches $3$ .

$\frac{2 {(lim_{x \to 3} x)}^{2} - 5 lim_{x \to 3} x - 1 \cdot 12}{6 - lim_{x \to 3} x - lim_{x \to 3} x^{2}}$

Step 9

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

$\frac{2 {(lim_{x \to 3} x)}^{2} - 5 lim_{x \to 3} x - 1 \cdot 12}{6 - lim_{x \to 3} x - {(lim_{x \to 3} x)}^{2}}$

Step 10

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{2 \cdot 3^{2} - 5 lim_{x \to 3} x - 1 \cdot 12}{6 - lim_{x \to 3} x - {(lim_{x \to 3} x)}^{2}}$

Step 10.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{2 \cdot 3^{2} - 5 \cdot 3 - 1 \cdot 12}{6 - lim_{x \to 3} x - {(lim_{x \to 3} x)}^{2}}$

Step 10.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{2 \cdot 3^{2} - 5 \cdot 3 - 1 \cdot 12}{6 - 3 - {(lim_{x \to 3} x)}^{2}}$

Step 10.4

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{2 \cdot 3^{2} - 5 \cdot 3 - 1 \cdot 12}{6 - 3 - 3^{2}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $3$ to the power of $2$ .

$\frac{2 \cdot 9 - 5 \cdot 3 - 1 \cdot 12}{6 - 3 - 3^{2}}$

Step 11.1.2

Multiply $2$ by $9$ .

$\frac{18 - 5 \cdot 3 - 1 \cdot 12}{6 - 3 - 3^{2}}$

Step 11.1.3

Multiply $- 5$ by $3$ .

$\frac{18 - 15 - 1 \cdot 12}{6 - 3 - 3^{2}}$

Step 11.1.4

Multiply $- 1$ by $12$ .

$\frac{18 - 15 - 12}{6 - 3 - 3^{2}}$

Step 11.1.5

Subtract $15$ from $18$ .

$\frac{3 - 12}{6 - 3 - 3^{2}}$

Step 11.1.6

Subtract $12$ from $3$ .

$\frac{- 9}{6 - 3 - 3^{2}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raise $3$ to the power of $2$ .

$\frac{- 9}{6 - 3 - 1 \cdot 9}$

Step 11.2.2

Multiply $- 1$ by $9$ .

$\frac{- 9}{6 - 3 - 9}$

Step 11.2.3

Subtract $3$ from $6$ .

$\frac{- 9}{3 - 9}$

Step 11.2.4

Subtract $9$ from $3$ .

$\frac{- 9}{- 6}$

Step 11.3

Cancel the common factor of $- 9$ and $- 6$ .

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Step 11.3.1

Factor $- 3$ out of $- 9$ .

$\frac{- 3 (3)}{- 6}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $- 3$ out of $- 6$ .

$\frac{- 3 \cdot 3}{- 3 \cdot 2}$

Step 11.3.2.2

Cancel the common factor.

$\frac{- 3 \cdot 3}{- 3 \cdot 2}$

Step 11.3.2.3

Rewrite the expression.

$\frac{3}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$