Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Move the exponent $2 - 4$ from $x^{2 - 4}$ outside the limit using the Limits Power Rule.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

Simplify the answer.

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Step 8.1

Move $2^{2 - 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 8.2

Simplify each term.

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Step 8.2.1

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

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

Step 8.2.2

Multiply $2$ by $2$ .

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

Step 8.2.3

Multiply $- 1$ by $8$ .

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

Step 8.3

Add $4$ and $4$ .

$(8 - 8) \cdot 2^{- (2 - 4)}$

Step 8.4

Subtract $8$ from $8$ .

$0 \cdot 2^{- (2 - 4)}$

Step 8.5

Subtract $4$ from $2$ .

$0 \cdot 2^{- - 2}$

Step 8.6

Multiply $- 1$ by $- 2$ .

$0 \cdot 2^{2}$

Step 8.7

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

$0 \cdot 4$

Step 8.8

Multiply $0$ by $4$ .

$0$