Calculus Examples

$lim_{x \to 0} \frac{x^{2} - 16}{x - 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x^{2} - 16}{lim_{x \to 0} x - 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x^{2} - lim_{x \to 0} 16}{lim_{x \to 0} x - 4}$

Step 3

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

$\frac{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 16}{lim_{x \to 0} x - 4}$

Step 4

Evaluate the limit of $16$ which is constant as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x)}^{2} - 1 \cdot 16}{lim_{x \to 0} x - 4}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x)}^{2} - 1 \cdot 16}{lim_{x \to 0} x - lim_{x \to 0} 4}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x)}^{2} - 1 \cdot 16}{lim_{x \to 0} x - 1 \cdot 4}$

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{2} - 1 \cdot 16}{lim_{x \to 0} x - 1 \cdot 4}$

Step 7.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{2} - 1 \cdot 16}{0 - 1 \cdot 4}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{0 - 1 \cdot 16}{0 - 1 \cdot 4}$

Step 8.1.2

Multiply $- 1$ by $16$ .

$\frac{0 - 16}{0 - 1 \cdot 4}$

Step 8.1.3

Subtract $16$ from $0$ .

$\frac{- 16}{0 - 1 \cdot 4}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $4$ .

$\frac{- 16}{0 - 4}$

Step 8.2.2

Subtract $4$ from $0$ .

$\frac{- 16}{- 4}$

Step 8.3

Divide $- 16$ by $- 4$ .

$4$