Calculus Examples

$\frac{lim_{x \to 3} x^{2} - 9}{5 x - 15}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 3} x^{2} - lim_{x \to 3} 9}{5 x - 15}$

Step 1.2

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

$\frac{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 9}{5 x - 15}$

Step 1.3

Evaluate the limit of $9$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 1 \cdot 9}{5 x - 15}$

Step 2

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

$\frac{3^{2} - 1 \cdot 9}{5 x - 15}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{9 - 1 \cdot 9}{5 x - 15}$

Step 3.1.2

Multiply $- 1$ by $9$ .

$\frac{9 - 9}{5 x - 15}$

Step 3.1.3

Subtract $9$ from $9$ .

$\frac{0}{5 x - 15}$

Step 3.2

Factor $5$ out of $5 x - 15$ .

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Step 3.2.1

Factor $5$ out of $5 x$ .

$\frac{0}{5 (x) - 15}$

Step 3.2.2

Factor $5$ out of $- 15$ .

$\frac{0}{5 x + 5 \cdot - 3}$

Step 3.2.3

Factor $5$ out of $5 x + 5 \cdot - 3$ .

$\frac{0}{5 (x - 3)}$

Step 3.3

Cancel the common factor of $0$ and $5$ .

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Step 3.3.1

Factor $5$ out of $0$ .

$\frac{5 (0)}{5 (x - 3)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Cancel the common factor.

$\frac{5 \cdot 0}{5 (x - 3)}$

Step 3.3.2.2

Rewrite the expression.

$\frac{0}{x - 3}$

Step 3.4

Divide $0$ by $x - 3$ .

$0$