Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

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

Step 3

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

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

Step 4

Move the exponent $2 - 16$ from $x^{2 - 16}$ outside the limit using the Limits Power Rule.

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

Step 5

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 5.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 6

Simplify the answer.

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Step 6.1

Move $4^{2 - 16}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$(4 - 1 \cdot 4) \cdot 4^{- (2 - 16)}$

Step 6.2

Multiply $- 1$ by $4$ .

$(4 - 4) \cdot 4^{- (2 - 16)}$

Step 6.3

Subtract $4$ from $4$ .

$0 \cdot 4^{- (2 - 16)}$

Step 6.4

Subtract $16$ from $2$ .

$0 \cdot 4^{- - 14}$

Step 6.5

Multiply $- 1$ by $- 14$ .

$0 \cdot 4^{14}$

Step 6.6

Raise $4$ to the power of $14$ .

$0 \cdot 268435456$

Step 6.7

Multiply $0$ by $268435456$ .

$0$