Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate the limit of $8$ which is constant as $x$ approaches $2$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

$\frac{2 \cdot 2^{2} - 1 \cdot 8}{2 + 2}$

Step 9

Simplify the answer.

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

Factor $- 1$ out of $- 1 \cdot 8$ .

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

Step 9.1.3

Factor $- 1$ out of $- (- 2 \cdot 2^{2}) - 1 (8)$ .

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

Step 9.1.4

Rewrite $- (- 2 \cdot 2^{2} + 8)$ as $- 1 (- 2 \cdot 2^{2} + 8)$ .

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

Step 9.1.5

Factor $2$ out of $- 1 (- 2 \cdot 2^{2} + 8)$ .

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

Step 9.1.6

Cancel the common factors.

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Step 9.1.6.1

Factor $2$ out of $2$ .

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

Step 9.1.6.2

Factor $2$ out of $2$ .

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

Step 9.1.6.3

Factor $2$ out of $2 \cdot 1 + 2 (1)$ .

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

Step 9.1.6.4

Cancel the common factor.

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

Step 9.1.6.5

Rewrite the expression.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

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

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

Step 9.2.2

Multiply $- 1$ by $4$ .

$\frac{- 1 (- 4 + 4)}{1 + 1}$

Step 9.2.3

Add $- 4$ and $4$ .

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

Step 9.3

Add $1$ and $1$ .

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

Step 9.4

Multiply $- 1$ by $0$ .

$\frac{0}{2}$

Step 9.5

Divide $0$ by $2$ .

$0$