Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $3^{3} - 3 y$ and $3^{2} - 1 \cdot 3$ .

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Step 9.1.1

Factor $- 1$ out of $3^{3}$ .

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

Step 9.1.2

Factor $- 1$ out of $- 3 y$ .

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

Step 9.1.3

Factor $- 1$ out of $- 1 (- 3^{3}) - (3 y)$ .

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

Step 9.1.4

Factor $3$ out of $- 1 (- 3^{3} + 3 y)$ .

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

Step 9.1.5

Cancel the common factors.

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Step 9.1.5.1

Factor $3$ out of $3^{2}$ .

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

Step 9.1.5.2

Factor $3$ out of $- 1 \cdot 3$ .

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

Step 9.1.5.3

Factor $3$ out of $3 \cdot 3 + 3 \cdot - 1$ .

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

Step 9.1.5.4

Cancel the common factor.

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

Step 9.1.5.5

Rewrite the expression.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

Raise $3$ to the power of $2$ .

$\frac{- 1 (- 1 \cdot 9 + y)}{3 - 1}$

Step 9.2.2

Multiply $- 1$ by $9$ .

$\frac{- 1 (- 9 + y)}{3 - 1}$

Step 9.3

Subtract $1$ from $3$ .

$\frac{- 1 (- 9 + y)}{2}$

Step 9.4

Move the negative in front of the fraction.

$- \frac{- 9 + y}{2}$

Step 9.5

Rewrite $- 9$ as $- 1 (9)$ .

$- \frac{- 1 (9) + y}{2}$

Step 9.6

Factor $- 1$ out of $y$ .

$- \frac{- 1 (9) - 1 (- y)}{2}$

Step 9.7

Factor $- 1$ out of $- 1 (9) - 1 (- y)$ .

$- \frac{- 1 (9 - y)}{2}$

Step 9.8

Move the negative in front of the fraction.

$- - \frac{9 - y}{2}$

Step 9.9

Multiply $- 1$ by $- 1$ .

$1 \frac{9 - y}{2}$

Step 9.10

Multiply $\frac{9 - y}{2}$ by $1$ .

$\frac{9 - y}{2}$