Calculus Examples

$\frac{lim_{x \to 2} 3 x^{2} - 4 x - 4}{2 x^{2} - 8}$

Step 1

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

$\frac{lim_{x \to 2} 3 x^{2} - lim_{x \to 2} 4 x - lim_{x \to 2} 4}{2 x^{2} - 8}$

Step 2

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

$\frac{3 lim_{x \to 2} x^{2} - lim_{x \to 2} 4 x - lim_{x \to 2} 4}{2 x^{2} - 8}$

Step 3

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 4 x - lim_{x \to 2} 4}{2 x^{2} - 8}$

Step 4

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x - lim_{x \to 2} 4}{2 x^{2} - 8}$

Step 5

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x - 1 \cdot 4}{2 x^{2} - 8}$

Step 6

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

Tap for more steps...

Step 6.1

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

$\frac{3 \cdot 2^{2} - 4 lim_{x \to 2} x - 1 \cdot 4}{2 x^{2} - 8}$

Step 6.2

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

$\frac{3 \cdot 2^{2} - 4 \cdot 2 - 1 \cdot 4}{2 x^{2} - 8}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Cancel the common factor of $3 \cdot 2^{2} - 4 \cdot 2 - 1 \cdot 4$ and $2 x^{2} - 8$ .

Tap for more steps...

Step 7.1.1

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

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

Step 7.1.2

Factor $- 1$ out of $- 4 \cdot 2$ .

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

Step 7.1.3

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

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

Step 7.1.4

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

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

Step 7.1.5

Factor $- 1$ out of $- (- 3 \cdot 2^{2} + 4 \cdot 2) - 1 (4)$ .

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

Step 7.1.6

Rewrite $- (- 3 \cdot 2^{2} + 4 \cdot 2 + 4)$ as $- 1 (- 3 \cdot 2^{2} + 4 \cdot 2 + 4)$ .

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

Step 7.1.7

Factor $2$ out of $- 1 (- 3 \cdot 2^{2} + 4 \cdot 2 + 4)$ .

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

Step 7.1.8

Cancel the common factors.

Tap for more steps...

Step 7.1.8.1

Factor $2$ out of $2 x^{2}$ .

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

Step 7.1.8.2

Factor $2$ out of $- 8$ .

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

Step 7.1.8.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 4$ .

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

Step 7.1.8.4

Cancel the common factor.

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

Step 7.1.8.5

Rewrite the expression.

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

Step 7.2

Simplify the numerator.

Tap for more steps...

Step 7.2.1

Multiply $- 3$ by $2$ .

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

Step 7.2.2

Multiply $2$ by $2$ .

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

Step 7.2.3

Add $- 6$ and $4$ .

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

Step 7.2.4

Add $- 2$ and $2$ .

$\frac{- 1 \cdot 0}{x^{2} - 4}$

Step 7.3

Simplify the denominator.

Tap for more steps...

Step 7.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{- 1 \cdot 0}{x^{2} - 2^{2}}$

Step 7.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{- 1 \cdot 0}{(x + 2) (x - 2)}$

Step 7.4

Multiply $- 1$ by $0$ .

$\frac{0}{(x + 2) (x - 2)}$

Step 7.5

Divide $0$ by $(x + 2) (x - 2)$ .

$0$