Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

Evaluate the limit of $15$ which is constant as $x$ approaches $3$ .

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

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 $x$ .

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $2$ by $3$ .

$\frac{9 + 6 - 1 \cdot 15}{x^{2} - 9}$

Step 6.1.3

Multiply $- 1$ by $15$ .

$\frac{9 + 6 - 15}{x^{2} - 9}$

Step 6.1.4

Add $9$ and $6$ .

$\frac{15 - 15}{x^{2} - 9}$

Step 6.1.5

Subtract $15$ from $15$ .

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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

Step 6.2.2

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 6.3

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

$0$