Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

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

Step 10.1.2

Multiply $- 1$ by $9$ .

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

Step 10.1.3

Subtract $9$ from $9$ .

$\frac{0}{3^{2} + 2 \cdot 3 - 1 \cdot 3}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

$\frac{0}{9 + 2 \cdot 3 - 1 \cdot 3}$

Step 10.2.2

Multiply $2$ by $3$ .

$\frac{0}{9 + 6 - 1 \cdot 3}$

Step 10.2.3

Multiply $- 1$ by $3$ .

$\frac{0}{9 + 6 - 3}$

Step 10.2.4

Add $9$ and $6$ .

$\frac{0}{15 - 3}$

Step 10.2.5

Subtract $3$ from $15$ .

$\frac{0}{12}$

Step 10.3

Divide $0$ by $12$ .

$0$