Calculus Examples

$\frac{lim_{x \to - 3} x^{2} - 5 x - 24}{x^{2} + 5 x + 6}$

Step 1

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

$\frac{lim_{x \to - 3} x^{2} - lim_{x \to - 3} 5 x - lim_{x \to - 3} 24}{x^{2} + 5 x + 6}$

Step 2

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

$\frac{{(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} 5 x - lim_{x \to - 3} 24}{x^{2} + 5 x + 6}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to - 3} x)}^{2} - 5 lim_{x \to - 3} x - lim_{x \to - 3} 24}{x^{2} + 5 x + 6}$

Step 4

Evaluate the limit of $24$ which is constant as $x$ approaches $- 3$ .

$\frac{{(lim_{x \to - 3} x)}^{2} - 5 lim_{x \to - 3} x - 1 \cdot 24}{x^{2} + 5 x + 6}$

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 $x$ .

$\frac{{(- 3)}^{2} - 5 lim_{x \to - 3} x - 1 \cdot 24}{x^{2} + 5 x + 6}$

Step 5.2

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

$\frac{{(- 3)}^{2} - 5 \cdot - 3 - 1 \cdot 24}{x^{2} + 5 x + 6}$

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 $2$ .

$\frac{9 - 5 \cdot - 3 - 1 \cdot 24}{x^{2} + 5 x + 6}$

Step 6.1.2

Multiply $- 5$ by $- 3$ .

$\frac{9 + 15 - 1 \cdot 24}{x^{2} + 5 x + 6}$

Step 6.1.3

Multiply $- 1$ by $24$ .

$\frac{9 + 15 - 24}{x^{2} + 5 x + 6}$

Step 6.1.4

Add $9$ and $15$ .

$\frac{24 - 24}{x^{2} + 5 x + 6}$

Step 6.1.5

Subtract $24$ from $24$ .

$\frac{0}{x^{2} + 5 x + 6}$

Step 6.2

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 6.2.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 6.2.2

Write the factored form using these integers.

$\frac{0}{(x + 2) (x + 3)}$

Step 6.3

Divide $0$ by $(x + 2) (x + 3)$ .

$0$