Calculus Examples

$lim_{x \to 8} \frac{x^{2} - 64}{x - 9}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x^{2} - 64}{lim_{x \to 8} x - 9}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x^{2} - lim_{x \to 8} 64}{lim_{x \to 8} x - 9}$

Step 3

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

$\frac{{(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 64}{lim_{x \to 8} x - 9}$

Step 4

Evaluate the limit of $64$ which is constant as $x$ approaches $8$ .

$\frac{{(lim_{x \to 8} x)}^{2} - 1 \cdot 64}{lim_{x \to 8} x - 9}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{{(lim_{x \to 8} x)}^{2} - 1 \cdot 64}{lim_{x \to 8} x - lim_{x \to 8} 9}$

Step 6

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$\frac{{(lim_{x \to 8} x)}^{2} - 1 \cdot 64}{lim_{x \to 8} x - 1 \cdot 9}$

Step 7

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8^{2} - 1 \cdot 64}{lim_{x \to 8} x - 1 \cdot 9}$

Step 7.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8^{2} - 1 \cdot 64}{8 - 1 \cdot 9}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $8$ to the power of $2$ .

$\frac{64 - 1 \cdot 64}{8 - 1 \cdot 9}$

Step 8.1.2

Multiply $- 1$ by $64$ .

$\frac{64 - 64}{8 - 1 \cdot 9}$

Step 8.1.3

Subtract $64$ from $64$ .

$\frac{0}{8 - 1 \cdot 9}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $9$ .

$\frac{0}{8 - 9}$

Step 8.2.2

Subtract $9$ from $8$ .

$\frac{0}{- 1}$

Step 8.3

Divide $0$ by $- 1$ .

$0$