Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

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

Step 11.1.2

Multiply $3$ by $4$ .

$\frac{12 - 4 \cdot 2 - 2 (3 \cdot 2^{2} - 4 \cdot 2)}{x - 2}$

Step 11.1.3

Multiply $- 4$ by $2$ .

$\frac{12 - 8 - 2 (3 \cdot 2^{2} - 4 \cdot 2)}{x - 2}$

Step 11.1.4

Simplify each term.

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Step 11.1.4.1

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

$\frac{12 - 8 - 2 (3 \cdot 4 - 4 \cdot 2)}{x - 2}$

Step 11.1.4.2

Multiply $3$ by $4$ .

$\frac{12 - 8 - 2 (12 - 4 \cdot 2)}{x - 2}$

Step 11.1.4.3

Multiply $- 4$ by $2$ .

$\frac{12 - 8 - 2 (12 - 8)}{x - 2}$

Step 11.1.5

Subtract $8$ from $12$ .

$\frac{12 - 8 - 2 \cdot 4}{x - 2}$

Step 11.1.6

Multiply $- 2$ by $4$ .

$\frac{12 - 8 - 8}{x - 2}$

Step 11.1.7

Subtract $8$ from $12$ .

$\frac{4 - 8}{x - 2}$

Step 11.1.8

Subtract $8$ from $4$ .

$\frac{- 4}{x - 2}$

Step 11.2

Move the negative in front of the fraction.

$- \frac{4}{x - 2}$