Calculus Examples

$\frac{lim_{x \to 3} 4 x^{2} - 7 x - 15}{x^{2} + 4 x - 21}$

Step 1

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

$\frac{lim_{x \to 3} 4 x^{2} - lim_{x \to 3} 7 x - lim_{x \to 3} 15}{x^{2} + 4 x - 21}$

Step 2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 3} x^{2} - lim_{x \to 3} 7 x - lim_{x \to 3} 15}{x^{2} + 4 x - 21}$

Step 3

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

$\frac{4 {(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 7 x - lim_{x \to 3} 15}{x^{2} + 4 x - 21}$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 {(lim_{x \to 3} x)}^{2} - 7 lim_{x \to 3} x - lim_{x \to 3} 15}{x^{2} + 4 x - 21}$

Step 5

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

$\frac{4 {(lim_{x \to 3} x)}^{2} - 7 lim_{x \to 3} x - 1 \cdot 15}{x^{2} + 4 x - 21}$

Step 6

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

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Step 6.1

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

$\frac{4 \cdot 3^{2} - 7 lim_{x \to 3} x - 1 \cdot 15}{x^{2} + 4 x - 21}$

Step 6.2

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

$\frac{4 \cdot 3^{2} - 7 \cdot 3 - 1 \cdot 15}{x^{2} + 4 x - 21}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{4 \cdot 9 - 7 \cdot 3 - 1 \cdot 15}{x^{2} + 4 x - 21}$

Step 7.1.2

Multiply $4$ by $9$ .

$\frac{36 - 7 \cdot 3 - 1 \cdot 15}{x^{2} + 4 x - 21}$

Step 7.1.3

Multiply $- 7$ by $3$ .

$\frac{36 - 21 - 1 \cdot 15}{x^{2} + 4 x - 21}$

Step 7.1.4

Multiply $- 1$ by $15$ .

$\frac{36 - 21 - 15}{x^{2} + 4 x - 21}$

Step 7.1.5

Subtract $21$ from $36$ .

$\frac{15 - 15}{x^{2} + 4 x - 21}$

Step 7.1.6

Subtract $15$ from $15$ .

$\frac{0}{x^{2} + 4 x - 21}$

Step 7.2

Factor $x^{2} + 4 x - 21$ using the AC method.

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Step 7.2.1

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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Step 7.2.2

Write the factored form using these integers.

$\frac{0}{(x - 3) (x + 7)}$

Step 7.3

Divide $0$ by $(x - 3) (x + 7)$ .

$0$