Calculus Examples

$lim_{x \to 3} \frac{x^{2 - 8}}{x - 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x^{2 - 8}}{lim_{x \to 3} x - 2}$

Step 2

Move the exponent $2 - 8$ from $x^{2 - 8}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 3} x)}^{2 - 8}}{lim_{x \to 3} x - 2}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{2 - 8}}{lim_{x \to 3} x - lim_{x \to 3} 2}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{2 - 8}}{lim_{x \to 3} x - 1 \cdot 2}$

Step 5

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{2 - 8}}{lim_{x \to 3} x - 1 \cdot 2}$

Step 5.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{2 - 8}}{3 - 1 \cdot 2}$

Step 6

Simplify the answer.

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Step 6.1

Move $3^{2 - 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{(3 - 1 \cdot 2) \cdot 3^{- (2 - 8)}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $- 1$ by $2$ .

$\frac{1}{(3 - 2) \cdot 3^{- (2 - 8)}}$

Step 6.2.2

Subtract $2$ from $3$ .

$\frac{1}{1 \cdot 3^{- (2 - 8)}}$

Step 6.2.3

Multiply $3^{- (2 - 8)}$ by $1$ .

$\frac{1}{3^{- (2 - 8)}}$

Step 6.2.4

Subtract $8$ from $2$ .

$\frac{1}{3^{- - 6}}$

Step 6.2.5

Multiply $- 1$ by $- 6$ .

$\frac{1}{3^{6}}$

Step 6.2.6

Raise $3$ to the power of $6$ .

$\frac{1}{729}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{729}$

Decimal Form:

$0.00137174 \dots$