Calculus Examples

$\frac{lim_{x \to 3} x^{3} - 3 x^{2} + 4 x - 12}{x^{4} - 3 x^{3} + x - 3}$

Step 1

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

$\frac{lim_{x \to 3} x^{3} - lim_{x \to 3} 3 x^{2} + lim_{x \to 3} 4 x - lim_{x \to 3} 12}{x^{4} - 3 x^{3} + x - 3}$

Step 2

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

$\frac{{(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 3 x^{2} + lim_{x \to 3} 4 x - lim_{x \to 3} 12}{x^{4} - 3 x^{3} + x - 3}$

Step 3

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

$\frac{{(lim_{x \to 3} x)}^{3} - 3 lim_{x \to 3} x^{2} + lim_{x \to 3} 4 x - lim_{x \to 3} 12}{x^{4} - 3 x^{3} + x - 3}$

Step 4

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

$\frac{{(lim_{x \to 3} x)}^{3} - 3 {(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 4 x - lim_{x \to 3} 12}{x^{4} - 3 x^{3} + x - 3}$

Step 5

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

$\frac{{(lim_{x \to 3} x)}^{3} - 3 {(lim_{x \to 3} x)}^{2} + 4 lim_{x \to 3} x - lim_{x \to 3} 12}{x^{4} - 3 x^{3} + x - 3}$

Step 6

Evaluate the limit of $12$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{3} - 3 {(lim_{x \to 3} x)}^{2} + 4 lim_{x \to 3} x - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 7

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

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Step 7.1

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

$\frac{3^{3} - 3 {(lim_{x \to 3} x)}^{2} + 4 lim_{x \to 3} x - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 7.2

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

$\frac{3^{3} - 3 \cdot 3^{2} + 4 lim_{x \to 3} x - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 7.3

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

$\frac{3^{3} - 3 \cdot 3^{2} + 4 \cdot 3 - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$\frac{27 - 3 \cdot 3^{2} + 4 \cdot 3 - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8.1.2

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

$\frac{27 - 3 \cdot 9 + 4 \cdot 3 - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8.1.3

Multiply $- 3$ by $9$ .

$\frac{27 - 27 + 4 \cdot 3 - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8.1.4

Multiply $4$ by $3$ .

$\frac{27 - 27 + 12 - 1 \cdot 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8.1.5

Multiply $- 1$ by $12$ .

$\frac{27 - 27 + 12 - 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8.1.6

Subtract $27$ from $27$ .

$\frac{0 + 12 - 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8.1.7

Add $0$ and $12$ .

$\frac{12 - 12}{x^{4} - 3 x^{3} + x - 3}$

Step 8.1.8

Subtract $12$ from $12$ .

$\frac{0}{x^{4} - 3 x^{3} + x - 3}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Factor out the greatest common factor from each group.

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Step 8.2.1.1

Group the first two terms and the last two terms.

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

Step 8.2.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{0}{x^{3} (x - 3) + 1 (x - 3)}$

Step 8.2.2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$\frac{0}{(x - 3) (x^{3} + 1)}$

Step 8.2.3

Rewrite $1$ as $1^{3}$ .

$\frac{0}{(x - 3) (x^{3} + 1^{3})}$

Step 8.2.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$\frac{0}{(x - 3) ((x + 1) (x^{2} - x \cdot 1 + 1^{2}))}$

Step 8.2.5

Simplify.

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Step 8.2.5.1

Multiply $- 1$ by $1$ .

$\frac{0}{(x - 3) ((x + 1) (x^{2} - x + 1^{2}))}$

Step 8.2.5.2

One to any power is one.

$\frac{0}{(x - 3) ((x + 1) (x^{2} - x + 1))}$

$\frac{0}{(x - 3) (x + 1) (x^{2} - x + 1)}$

Step 8.3

Divide $0$ by $(x - 3) (x + 1) (x^{2} - x + 1)$ .

$0$