Calculus Examples

$lim_{x \to 2} \frac{3 x^{2} - 6 x}{4 - 3 x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} 3 x^{2} - 6 x}{lim_{x \to 2} 4 - 3 x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} 3 x^{2} - lim_{x \to 2} 6 x}{lim_{x \to 2} 4 - 3 x}$

Step 3

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

$\frac{3 lim_{x \to 2} x^{2} - lim_{x \to 2} 6 x}{lim_{x \to 2} 4 - 3 x}$

Step 4

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 6 x}{lim_{x \to 2} 4 - 3 x}$

Step 5

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - 6 lim_{x \to 2} x}{lim_{x \to 2} 4 - 3 x}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 6 lim_{x \to 2} x}{lim_{x \to 2} 4 - lim_{x \to 2} 3 x}$

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 6 lim_{x \to 2} x}{4 - lim_{x \to 2} 3 x}$

Step 8

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - 6 lim_{x \to 2} x}{4 - 3 lim_{x \to 2} x}$

Step 9

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{3 \cdot 2^{2} - 6 lim_{x \to 2} x}{4 - 3 lim_{x \to 2} x}$

Step 9.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{3 \cdot 2^{2} - 6 \cdot 2}{4 - 3 lim_{x \to 2} x}$

Step 9.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{3 \cdot 2^{2} - 6 \cdot 2}{4 - 3 \cdot 2}$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $3 \cdot 2^{2} - 6 \cdot 2$ and $4 - 3 \cdot 2$ .

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Step 10.1.1

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{3 \cdot 2^{2} - 6 \cdot 2}{- 1 (- 4) - 3 \cdot 2}$

Step 10.1.2

Factor $- 1$ out of $- 3 \cdot 2$ .

$\frac{3 \cdot 2^{2} - 6 \cdot 2}{- 1 (- 4) - (3 \cdot 2)}$

Step 10.1.3

Factor $- 1$ out of $- 1 (- 4) - (3 \cdot 2)$ .

$\frac{3 \cdot 2^{2} - 6 \cdot 2}{- 1 (- 4 + 3 \cdot 2)}$

Step 10.1.4

Reorder terms.

$\frac{3 \cdot 2^{2} - 6 \cdot 2}{- 1 (2 \cdot 3 - 4)}$

Step 10.1.5

Factor $2$ out of $3 \cdot 2^{2}$ .

$\frac{2 (3 \cdot 2) - 6 \cdot 2}{- 1 (2 \cdot 3 - 4)}$

Step 10.1.6

Factor $2$ out of $- 6 \cdot 2$ .

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

Step 10.1.7

Factor $2$ out of $2 (3 \cdot 2) + 2 (- 3 \cdot 2)$ .

$\frac{2 (3 \cdot 2 - 3 \cdot 2)}{- 1 (2 \cdot 3 - 4)}$

Step 10.1.8

Cancel the common factors.

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Step 10.1.8.1

Factor $2$ out of $- 1 (2 \cdot 3 - 4)$ .

$\frac{2 (3 \cdot 2 - 3 \cdot 2)}{2 (- 1 (3 - 2))}$

Step 10.1.8.2

Cancel the common factor.

$\frac{2 (3 \cdot 2 - 3 \cdot 2)}{2 (- 1 (3 - 2))}$

Step 10.1.8.3

Rewrite the expression.

$\frac{3 \cdot 2 - 3 \cdot 2}{- 1 (3 - 2)}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Multiply $3$ by $2$ .

$\frac{6 - 3 \cdot 2}{- 1 (3 - 2)}$

Step 10.2.2

Multiply $- 3$ by $2$ .

$\frac{6 - 6}{- 1 (3 - 2)}$

Step 10.2.3

Subtract $6$ from $6$ .

$\frac{0}{- 1 (3 - 2)}$

Step 10.3

Subtract $2$ from $3$ .

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

Step 10.4

Multiply $- 1$ by $1$ .

$\frac{0}{- 1}$

Step 10.5

Divide $0$ by $- 1$ .

$0$