Calculus Examples

$\frac{lim_{x \to - 3} x^{2} + x - 6}{x^{2} - 9}$

Step 1

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

$\frac{lim_{x \to - 3} x^{2} + lim_{x \to - 3} x - lim_{x \to - 3} 6}{x^{2} - 9}$

Step 2

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

$\frac{{(lim_{x \to - 3} x)}^{2} + lim_{x \to - 3} x - lim_{x \to - 3} 6}{x^{2} - 9}$

Step 3

Evaluate the limit of $6$ which is constant as $x$ approaches $- 3$ .

$\frac{{(lim_{x \to - 3} x)}^{2} + lim_{x \to - 3} x - 1 \cdot 6}{x^{2} - 9}$

Step 4

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

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Step 4.1

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

$\frac{{(- 3)}^{2} + lim_{x \to - 3} x - 1 \cdot 6}{x^{2} - 9}$

Step 4.2

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

$\frac{{(- 3)}^{2} - 3 - 1 \cdot 6}{x^{2} - 9}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{9 - 3 - 1 \cdot 6}{x^{2} - 9}$

Step 5.1.2

Multiply $- 1$ by $6$ .

$\frac{9 - 3 - 6}{x^{2} - 9}$

Step 5.1.3

Subtract $3$ from $9$ .

$\frac{6 - 6}{x^{2} - 9}$

Step 5.1.4

Subtract $6$ from $6$ .

$\frac{0}{x^{2} - 9}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{0}{x^{2} - 3^{2}}$

Step 5.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 5.3

Divide $0$ by $(x + 3) (x - 3)$ .

$0$