Calculus Examples

$\frac{lim_{x \to 3} x^{2} - 3 x}{x^{2} + x - 12}$

Step 1

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

$\frac{lim_{x \to 3} x^{2} - lim_{x \to 3} 3 x}{x^{2} + x - 12}$

Step 2

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

$\frac{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 3 x}{x^{2} + x - 12}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 3 lim_{x \to 3} x}{x^{2} + x - 12}$

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 $x$ .

$\frac{3^{2} - 3 lim_{x \to 3} x}{x^{2} + x - 12}$

Step 4.2

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

$\frac{3^{2} - 3 \cdot 3}{x^{2} + x - 12}$

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 $2$ .

$\frac{9 - 3 \cdot 3}{x^{2} + x - 12}$

Step 5.1.2

Multiply $- 3$ by $3$ .

$\frac{9 - 9}{x^{2} + x - 12}$

Step 5.1.3

Subtract $9$ from $9$ .

$\frac{0}{x^{2} + x - 12}$

Step 5.2

Factor $x^{2} + x - 12$ using the AC method.

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Step 5.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 5.2.2

Write the factored form using these integers.

$\frac{0}{(x - 3) (x + 4)}$

Step 5.3

Divide $0$ by $(x - 3) (x + 4)$ .

$0$