Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

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 $x$ .

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

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

Step 9.1.2

Multiply $- 1$ by $4$ .

$\frac{4 - 4}{2^{2} + 4}$

Step 9.1.3

Subtract $4$ from $4$ .

$\frac{0}{2^{2} + 4}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$\frac{0}{4 + 4}$

Step 9.2.2

Add $4$ and $4$ .

$\frac{0}{8}$

Step 9.3

Divide $0$ by $8$ .

$0$