Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $3^{2} - 2 \cdot 3 - 1 \cdot 15$ and $3 + 3$ .

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Step 9.1.1

Factor $- 1$ out of $3^{2}$ .

$\frac{- 1 (- 3^{2}) - 2 \cdot 3 - 1 \cdot 15}{3 + 3}$

Step 9.1.2

Factor $- 1$ out of $- 2 \cdot 3$ .

$\frac{- 1 (- 3^{2}) - (2 \cdot 3) - 1 \cdot 15}{3 + 3}$

Step 9.1.3

Factor $- 1$ out of $- 1 (- 3^{2}) - (2 \cdot 3)$ .

$\frac{- 1 (- 3^{2} + 2 \cdot 3) - 1 \cdot 15}{3 + 3}$

Step 9.1.4

Factor $- 1$ out of $- 1 \cdot 15$ .

$\frac{- 1 (- 3^{2} + 2 \cdot 3) - 1 (15)}{3 + 3}$

Step 9.1.5

Factor $- 1$ out of $- 1 (- 3^{2} + 2 \cdot 3) - 1 (15)$ .

$\frac{- 1 (- 3^{2} + 2 \cdot 3 + 15)}{3 + 3}$

Step 9.1.6

Factor $3$ out of $- 1 (- 3^{2} + 2 \cdot 3 + 15)$ .

$\frac{3 (- 1 (- 1 \cdot 3 + 2 + 5))}{3 + 3}$

Step 9.1.7

Cancel the common factors.

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Step 9.1.7.1

Factor $3$ out of $3$ .

$\frac{3 (- 1 (- 1 \cdot 3 + 2 + 5))}{3 \cdot 1 + 3}$

Step 9.1.7.2

Factor $3$ out of $3$ .

$\frac{3 (- 1 (- 1 \cdot 3 + 2 + 5))}{3 \cdot 1 + 3 (1)}$

Step 9.1.7.3

Factor $3$ out of $3 \cdot 1 + 3 (1)$ .

$\frac{3 (- 1 (- 1 \cdot 3 + 2 + 5))}{3 \cdot (1 + 1)}$

Step 9.1.7.4

Cancel the common factor.

$\frac{3 (- 1 (- 1 \cdot 3 + 2 + 5))}{3 \cdot (1 + 1)}$

Step 9.1.7.5

Rewrite the expression.

$\frac{- 1 (- 1 \cdot 3 + 2 + 5)}{1 + 1}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

Multiply $- 1$ by $3$ .

$\frac{- 1 (- 3 + 2 + 5)}{1 + 1}$

Step 9.2.2

Add $- 3$ and $2$ .

$\frac{- 1 (- 1 + 5)}{1 + 1}$

Step 9.2.3

Add $- 1$ and $5$ .

$\frac{- 1 \cdot 4}{1 + 1}$

Step 9.3

Add $1$ and $1$ .

$\frac{- 1 \cdot 4}{2}$

Step 9.4

Multiply $- 1$ by $4$ .

$\frac{- 4}{2}$

Step 9.5

Divide $- 4$ by $2$ .

$- 2$