Calculus Examples

$\frac{lim_{x \to 3} x^{3} + x^{2} - 21 x + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 1

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

$\frac{lim_{x \to 3} x^{3} + lim_{x \to 3} x^{2} - lim_{x \to 3} 21 x + lim_{x \to 3} 45}{x^{3} + 6 x^{2} + 9 x}$

Step 2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 3} x)}^{3} + lim_{x \to 3} x^{2} - lim_{x \to 3} 21 x + lim_{x \to 3} 45}{x^{3} + 6 x^{2} + 9 x}$

Step 3

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

$\frac{{(lim_{x \to 3} x)}^{3} + {(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 21 x + lim_{x \to 3} 45}{x^{3} + 6 x^{2} + 9 x}$

Step 4

Move the term $21$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{3} + {(lim_{x \to 3} x)}^{2} - 21 lim_{x \to 3} x + lim_{x \to 3} 45}{x^{3} + 6 x^{2} + 9 x}$

Step 5

Evaluate the limit of $45$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{3} + {(lim_{x \to 3} x)}^{2} - 21 lim_{x \to 3} x + 45}{x^{3} + 6 x^{2} + 9 x}$

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 $x$ .

$\frac{3^{3} + {(lim_{x \to 3} x)}^{2} - 21 lim_{x \to 3} x + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 6.2

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

$\frac{3^{3} + 3^{2} - 21 lim_{x \to 3} x + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 6.3

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

$\frac{3^{3} + 3^{2} - 21 \cdot 3 + 45}{x^{3} + 6 x^{2} + 9 x}$

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 $3$ .

$\frac{27 + 3^{2} - 21 \cdot 3 + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 7.1.2

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

$\frac{27 + 9 - 21 \cdot 3 + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 7.1.3

Multiply $- 21$ by $3$ .

$\frac{27 + 9 - 63 + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 7.1.4

Add $27$ and $9$ .

$\frac{36 - 63 + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 7.1.5

Subtract $63$ from $36$ .

$\frac{- 27 + 45}{x^{3} + 6 x^{2} + 9 x}$

Step 7.1.6

Add $- 27$ and $45$ .

$\frac{18}{x^{3} + 6 x^{2} + 9 x}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Factor $x$ out of $x^{3} + 6 x^{2} + 9 x$ .

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Step 7.2.1.1

Factor $x$ out of $x^{3}$ .

$\frac{18}{x \cdot x^{2} + 6 x^{2} + 9 x}$

Step 7.2.1.2

Factor $x$ out of $6 x^{2}$ .

$\frac{18}{x \cdot x^{2} + x (6 x) + 9 x}$

Step 7.2.1.3

Factor $x$ out of $9 x$ .

$\frac{18}{x \cdot x^{2} + x (6 x) + x \cdot 9}$

Step 7.2.1.4

Factor $x$ out of $x \cdot x^{2} + x (6 x)$ .

$\frac{18}{x (x^{2} + 6 x) + x \cdot 9}$

Step 7.2.1.5

Factor $x$ out of $x (x^{2} + 6 x) + x \cdot 9$ .

$\frac{18}{x (x^{2} + 6 x + 9)}$

Step 7.2.2

Factor using the perfect square rule.

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Step 7.2.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{18}{x (x^{2} + 6 x + 3^{2})}$

Step 7.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 7.2.2.3

Rewrite the polynomial.

$\frac{18}{x (x^{2} + 2 \cdot x \cdot 3 + 3^{2})}$

Step 7.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{18}{x {(x + 3)}^{2}}$