Calculus Examples

$lim_{x \to - 3} \frac{3 x^{2} + 5 x - 12}{8 x - 9}$

Step 1

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

$\frac{lim_{x \to - 3} 3 x^{2} + 5 x - 12}{lim_{x \to - 3} 8 x - 9}$

Step 2

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

$\frac{lim_{x \to - 3} 3 x^{2} + lim_{x \to - 3} 5 x - lim_{x \to - 3} 12}{lim_{x \to - 3} 8 x - 9}$

Step 3

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

$\frac{3 lim_{x \to - 3} x^{2} + lim_{x \to - 3} 5 x - lim_{x \to - 3} 12}{lim_{x \to - 3} 8 x - 9}$

Step 4

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

$\frac{3 {(lim_{x \to - 3} x)}^{2} + lim_{x \to - 3} 5 x - lim_{x \to - 3} 12}{lim_{x \to - 3} 8 x - 9}$

Step 5

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

$\frac{3 {(lim_{x \to - 3} x)}^{2} + 5 lim_{x \to - 3} x - lim_{x \to - 3} 12}{lim_{x \to - 3} 8 x - 9}$

Step 6

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

$\frac{3 {(lim_{x \to - 3} x)}^{2} + 5 lim_{x \to - 3} x - 1 \cdot 12}{lim_{x \to - 3} 8 x - 9}$

Step 7

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

$\frac{3 {(lim_{x \to - 3} x)}^{2} + 5 lim_{x \to - 3} x - 1 \cdot 12}{lim_{x \to - 3} 8 x - lim_{x \to - 3} 9}$

Step 8

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to - 3} x)}^{2} + 5 lim_{x \to - 3} x - 1 \cdot 12}{8 lim_{x \to - 3} x - lim_{x \to - 3} 9}$

Step 9

Evaluate the limit of $9$ which is constant as $x$ approaches $- 3$ .

$\frac{3 {(lim_{x \to - 3} x)}^{2} + 5 lim_{x \to - 3} x - 1 \cdot 12}{8 lim_{x \to - 3} x - 1 \cdot 9}$

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 $x$ .

$\frac{3 {(- 3)}^{2} + 5 lim_{x \to - 3} x - 1 \cdot 12}{8 lim_{x \to - 3} x - 1 \cdot 9}$

Step 10.2

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

$\frac{3 {(- 3)}^{2} + 5 \cdot - 3 - 1 \cdot 12}{8 lim_{x \to - 3} x - 1 \cdot 9}$

Step 10.3

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

$\frac{3 {(- 3)}^{2} + 5 \cdot - 3 - 1 \cdot 12}{8 \cdot - 3 - 1 \cdot 9}$

Step 11

Simplify the answer.

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Step 11.1

Cancel the common factor of $3 {(- 3)}^{2} + 5 \cdot - 3 - 1 \cdot 12$ and $8 \cdot - 3 - 1 \cdot 9$ .

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Step 11.1.1

Factor $- 1$ out of $3 {(- 3)}^{2}$ .

$\frac{- (- 3 {(- 3)}^{2}) + 5 \cdot - 3 - 1 \cdot 12}{8 \cdot - 3 - 1 \cdot 9}$

Step 11.1.2

Factor $- 1$ out of $5 \cdot - 3$ .

$\frac{- (- 3 {(- 3)}^{2}) - (- 5 \cdot - 3) - 1 \cdot 12}{8 \cdot - 3 - 1 \cdot 9}$

Step 11.1.3

Factor $- 1$ out of $- (- 3 {(- 3)}^{2}) - (- 5 \cdot - 3)$ .

$\frac{- (- 3 {(- 3)}^{2} - 5 \cdot - 3) - 1 \cdot 12}{8 \cdot - 3 - 1 \cdot 9}$

Step 11.1.4

Factor $- 1$ out of $- 1 \cdot 12$ .

$\frac{- (- 3 {(- 3)}^{2} - 5 \cdot - 3) - 1 (12)}{8 \cdot - 3 - 1 \cdot 9}$

Step 11.1.5

Factor $- 1$ out of $- (- 3 {(- 3)}^{2} - 5 \cdot - 3) - 1 (12)$ .

$\frac{- (- 3 {(- 3)}^{2} - 5 \cdot - 3 + 12)}{8 \cdot - 3 - 1 \cdot 9}$

Step 11.1.6

Rewrite $- (- 3 {(- 3)}^{2} - 5 \cdot - 3 + 12)$ as $- 1 (- 3 {(- 3)}^{2} - 5 \cdot - 3 + 12)$ .

$\frac{- 1 (- 3 {(- 3)}^{2} - 5 \cdot - 3 + 12)}{8 \cdot - 3 - 1 \cdot 9}$

Step 11.1.7

Reorder terms.

$\frac{- 1 (- 3 {(- 3)}^{2} - 5 \cdot - 3 + 12)}{- 1 \cdot 9 - 3 \cdot 8}$

Step 11.1.8

Factor $3$ out of $- 1 (- 3 {(- 3)}^{2} - 5 \cdot - 3 + 12)$ .

$\frac{3 (- 1 (- {(- 3)}^{2} - 5 \cdot - 1 + 4))}{- 1 \cdot 9 - 3 \cdot 8}$

Step 11.1.9

Cancel the common factors.

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Step 11.1.9.1

Factor $3$ out of $- 1 \cdot 9$ .

$\frac{3 (- 1 (- {(- 3)}^{2} - 5 \cdot - 1 + 4))}{3 (- 1 \cdot 3) - 3 \cdot 8}$

Step 11.1.9.2

Factor $3$ out of $- 3 \cdot 8$ .

$\frac{3 (- 1 (- {(- 3)}^{2} - 5 \cdot - 1 + 4))}{3 (- 1 \cdot 3) + 3 (- 1 \cdot 8)}$

Step 11.1.9.3

Factor $3$ out of $3 (- 1 \cdot 3) + 3 (- 1 \cdot 8)$ .

$\frac{3 (- 1 (- {(- 3)}^{2} - 5 \cdot - 1 + 4))}{3 (- 1 \cdot 3 - 1 \cdot 8)}$

Step 11.1.9.4

Cancel the common factor.

$\frac{3 (- 1 (- {(- 3)}^{2} - 5 \cdot - 1 + 4))}{3 (- 1 \cdot 3 - 1 \cdot 8)}$

Step 11.1.9.5

Rewrite the expression.

$\frac{- 1 (- {(- 3)}^{2} - 5 \cdot - 1 + 4)}{- 1 \cdot 3 - 1 \cdot 8}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

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

$\frac{- 1 (- 1 \cdot 9 - 5 \cdot - 1 + 4)}{- 1 \cdot 3 - 1 \cdot 8}$

Step 11.2.2

Multiply $- 1$ by $9$ .

$\frac{- 1 (- 9 - 5 \cdot - 1 + 4)}{- 1 \cdot 3 - 1 \cdot 8}$

Step 11.2.3

Multiply $- 5$ by $- 1$ .

$\frac{- 1 (- 9 + 5 + 4)}{- 1 \cdot 3 - 1 \cdot 8}$

Step 11.2.4

Add $- 9$ and $5$ .

$\frac{- 1 (- 4 + 4)}{- 1 \cdot 3 - 1 \cdot 8}$

Step 11.2.5

Add $- 4$ and $4$ .

$\frac{- 1 \cdot 0}{- 1 \cdot 3 - 1 \cdot 8}$

Step 11.3

Simplify the denominator.

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Step 11.3.1

Multiply $- 1$ by $3$ .

$\frac{- 1 \cdot 0}{- 3 - 1 \cdot 8}$

Step 11.3.2

Multiply $- 1$ by $8$ .

$\frac{- 1 \cdot 0}{- 3 - 8}$

Step 11.3.3

Subtract $8$ from $- 3$ .

$\frac{- 1 \cdot 0}{- 11}$

Step 11.4

Multiply $- 1$ by $0$ .

$\frac{0}{- 11}$

Step 11.5

Divide $0$ by $- 11$ .

$0$