Calculus Examples

$lim_{x \to - 2} \frac{x - 4}{6 x^{2} - 8 x + 4}$

Step 1

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

$\frac{lim_{x \to - 2} x - 4}{lim_{x \to - 2} 6 x^{2} - 8 x + 4}$

Step 2

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

$\frac{lim_{x \to - 2} x - lim_{x \to - 2} 4}{lim_{x \to - 2} 6 x^{2} - 8 x + 4}$

Step 3

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

$\frac{lim_{x \to - 2} x - 1 \cdot 4}{lim_{x \to - 2} 6 x^{2} - 8 x + 4}$

Step 4

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

$\frac{lim_{x \to - 2} x - 1 \cdot 4}{lim_{x \to - 2} 6 x^{2} - lim_{x \to - 2} 8 x + lim_{x \to - 2} 4}$

Step 5

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

$\frac{lim_{x \to - 2} x - 1 \cdot 4}{6 lim_{x \to - 2} x^{2} - lim_{x \to - 2} 8 x + lim_{x \to - 2} 4}$

Step 6

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

$\frac{lim_{x \to - 2} x - 1 \cdot 4}{6 {(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 8 x + lim_{x \to - 2} 4}$

Step 7

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

$\frac{lim_{x \to - 2} x - 1 \cdot 4}{6 {(lim_{x \to - 2} x)}^{2} - 8 lim_{x \to - 2} x + lim_{x \to - 2} 4}$

Step 8

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

$\frac{lim_{x \to - 2} x - 1 \cdot 4}{6 {(lim_{x \to - 2} x)}^{2} - 8 lim_{x \to - 2} x + 4}$

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 $x$ .

$\frac{- 2 - 1 \cdot 4}{6 {(lim_{x \to - 2} x)}^{2} - 8 lim_{x \to - 2} x + 4}$

Step 9.2

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

$\frac{- 2 - 1 \cdot 4}{6 {(- 2)}^{2} - 8 lim_{x \to - 2} x + 4}$

Step 9.3

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

$\frac{- 2 - 1 \cdot 4}{6 {(- 2)}^{2} - 8 \cdot - 2 + 4}$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $- 2 - 1 \cdot 4$ and $6 {(- 2)}^{2} - 8 \cdot - 2 + 4$ .

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Step 10.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (2) - 1 \cdot 4}{6 {(- 2)}^{2} - 8 \cdot - 2 + 4}$

Step 10.1.2

Factor $- 1$ out of $- 1 \cdot 4$ .

$\frac{- 1 (2) - 1 (4)}{6 {(- 2)}^{2} - 8 \cdot - 2 + 4}$

Step 10.1.3

Factor $- 1$ out of $- 1 (2) - 1 (4)$ .

$\frac{- 1 (2 + 4)}{6 {(- 2)}^{2} - 8 \cdot - 2 + 4}$

Step 10.1.4

Factor $2$ out of $- 1 (2 + 4)$ .

$\frac{2 (- 1 (1 + 2))}{6 {(- 2)}^{2} - 8 \cdot - 2 + 4}$

Step 10.1.5

Cancel the common factors.

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Step 10.1.5.1

Factor $2$ out of $6 {(- 2)}^{2}$ .

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

Step 10.1.5.2

Factor $2$ out of $- 8 \cdot - 2$ .

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

Step 10.1.5.3

Factor $2$ out of $2 (3 {(- 2)}^{2}) + 2 (- 4 \cdot - 2)$ .

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

Step 10.1.5.4

Factor $2$ out of $4$ .

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

Step 10.1.5.5

Factor $2$ out of $2 (3 {(- 2)}^{2} - 4 \cdot - 2) + 2 (2)$ .

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

Step 10.1.5.6

Cancel the common factor.

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

Step 10.1.5.7

Rewrite the expression.

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

Step 10.2

Add $1$ and $2$ .

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

Step 10.3

Simplify the denominator.

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Step 10.3.1

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

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

Step 10.3.2

Multiply $3$ by $4$ .

$\frac{- 1 \cdot 3}{12 - 4 \cdot - 2 + 2}$

Step 10.3.3

Multiply $- 4$ by $- 2$ .

$\frac{- 1 \cdot 3}{12 + 8 + 2}$

Step 10.3.4

Add $12$ and $8$ .

$\frac{- 1 \cdot 3}{20 + 2}$

Step 10.3.5

Add $20$ and $2$ .

$\frac{- 1 \cdot 3}{22}$

Step 10.4

Multiply $- 1$ by $3$ .

$\frac{- 3}{22}$

Step 10.5

Move the negative in front of the fraction.

$- \frac{3}{22}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{22}$

Decimal Form:

$- 0.1\overline{36}$