Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

$\frac{{(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 9 x^{2} + lim_{x \to 3} 18 x}{lim_{x \to 3} x^{3} + 6 x^{2} + 9 x}$

Step 4

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{3} - 9 lim_{x \to 3} x^{2} + lim_{x \to 3} 18 x}{lim_{x \to 3} x^{3} + 6 x^{2} + 9 x}$

Step 5

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

$\frac{{(lim_{x \to 3} x)}^{3} - 9 {(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 18 x}{lim_{x \to 3} x^{3} + 6 x^{2} + 9 x}$

Step 6

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{3} - 9 {(lim_{x \to 3} x)}^{2} + 18 lim_{x \to 3} x}{lim_{x \to 3} x^{3} + 6 x^{2} + 9 x}$

Step 7

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

$\frac{{(lim_{x \to 3} x)}^{3} - 9 {(lim_{x \to 3} x)}^{2} + 18 lim_{x \to 3} x}{lim_{x \to 3} x^{3} + lim_{x \to 3} 6 x^{2} + lim_{x \to 3} 9 x}$

Step 8

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

$\frac{{(lim_{x \to 3} x)}^{3} - 9 {(lim_{x \to 3} x)}^{2} + 18 lim_{x \to 3} x}{{(lim_{x \to 3} x)}^{3} + lim_{x \to 3} 6 x^{2} + lim_{x \to 3} 9 x}$

Step 9

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{3} - 9 {(lim_{x \to 3} x)}^{2} + 18 lim_{x \to 3} x}{{(lim_{x \to 3} x)}^{3} + 6 lim_{x \to 3} x^{2} + lim_{x \to 3} 9 x}$

Step 10

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

$\frac{{(lim_{x \to 3} x)}^{3} - 9 {(lim_{x \to 3} x)}^{2} + 18 lim_{x \to 3} x}{{(lim_{x \to 3} x)}^{3} + 6 {(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 9 x}$

Step 11

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{3} - 9 {(lim_{x \to 3} x)}^{2} + 18 lim_{x \to 3} x}{{(lim_{x \to 3} x)}^{3} + 6 {(lim_{x \to 3} x)}^{2} + 9 lim_{x \to 3} x}$

Step 12

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

Tap for more steps...

Step 12.1

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

$\frac{3^{3} - 9 {(lim_{x \to 3} x)}^{2} + 18 lim_{x \to 3} x}{{(lim_{x \to 3} x)}^{3} + 6 {(lim_{x \to 3} x)}^{2} + 9 lim_{x \to 3} x}$

Step 12.2

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

$\frac{3^{3} - 9 \cdot 3^{2} + 18 lim_{x \to 3} x}{{(lim_{x \to 3} x)}^{3} + 6 {(lim_{x \to 3} x)}^{2} + 9 lim_{x \to 3} x}$

Step 12.3

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

$\frac{3^{3} - 9 \cdot 3^{2} + 18 \cdot 3}{{(lim_{x \to 3} x)}^{3} + 6 {(lim_{x \to 3} x)}^{2} + 9 lim_{x \to 3} x}$

Step 12.4

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

$\frac{3^{3} - 9 \cdot 3^{2} + 18 \cdot 3}{3^{3} + 6 {(lim_{x \to 3} x)}^{2} + 9 lim_{x \to 3} x}$

Step 12.5

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

$\frac{3^{3} - 9 \cdot 3^{2} + 18 \cdot 3}{3^{3} + 6 \cdot 3^{2} + 9 lim_{x \to 3} x}$

Step 12.6

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

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

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

Step 13

Simplify the answer.

Tap for more steps...

Step 13.1

Cancel the common factor of $3^{3} - 9 \cdot 3^{2} + 18 \cdot 3$ and $3^{3} + 6 \cdot 3^{2} + 9 \cdot 3$ .

Tap for more steps...

Step 13.1.1

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

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

Step 13.1.2

Factor $3$ out of $- 9 \cdot 3^{2}$ .

$\frac{3 \cdot 3^{2} + 3 (- 3 \cdot 3^{2}) + 18 \cdot 3}{3^{3} + 6 \cdot 3^{2} + 9 \cdot 3}$

Step 13.1.3

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

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2}) + 18 \cdot 3}{3^{3} + 6 \cdot 3^{2} + 9 \cdot 3}$

Step 13.1.4

Factor $3$ out of $18 \cdot 3$ .

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2}) + 3 (6 \cdot 3)}{3^{3} + 6 \cdot 3^{2} + 9 \cdot 3}$

Step 13.1.5

Factor $3$ out of $3 \cdot (3^{2} - 3 \cdot 3^{2}) + 3 (6 \cdot 3)$ .

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2} + 6 \cdot 3)}{3^{3} + 6 \cdot 3^{2} + 9 \cdot 3}$

Step 13.1.6

Cancel the common factors.

Tap for more steps...

Step 13.1.6.1

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

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2} + 6 \cdot 3)}{3 \cdot 3^{2} + 6 \cdot 3^{2} + 9 \cdot 3}$

Step 13.1.6.2

Factor $3$ out of $6 \cdot 3^{2}$ .

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2} + 6 \cdot 3)}{3 \cdot 3^{2} + 3 (2 \cdot 3^{2}) + 9 \cdot 3}$

Step 13.1.6.3

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

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2} + 6 \cdot 3)}{3 \cdot (3^{2} + 2 \cdot 3^{2}) + 9 \cdot 3}$

Step 13.1.6.4

Factor $3$ out of $9 \cdot 3$ .

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2} + 6 \cdot 3)}{3 \cdot (3^{2} + 2 \cdot 3^{2}) + 3 (3 \cdot 3)}$

Step 13.1.6.5

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

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2} + 6 \cdot 3)}{3 \cdot (3^{2} + 2 \cdot 3^{2} + 3 \cdot 3)}$

Step 13.1.6.6

Cancel the common factor.

$\frac{3 \cdot (3^{2} - 3 \cdot 3^{2} + 6 \cdot 3)}{3 \cdot (3^{2} + 2 \cdot 3^{2} + 3 \cdot 3)}$

Step 13.1.6.7

Rewrite the expression.

$\frac{3^{2} - 3 \cdot 3^{2} + 6 \cdot 3}{3^{2} + 2 \cdot 3^{2} + 3 \cdot 3}$

$\frac{3^{2} - 3 \cdot 3^{2} + 6 \cdot 3}{3^{2} + 2 \cdot 3^{2} + 3 \cdot 3}$

$\frac{3^{2} - 3 \cdot 3^{2} + 6 \cdot 3}{3^{2} + 2 \cdot 3^{2} + 3 \cdot 3}$

Step 13.2

Cancel the common factor of $3^{2} - 3 \cdot 3^{2} + 6 \cdot 3$ and $3^{2} + 2 \cdot 3^{2} + 3 \cdot 3$ .

Tap for more steps...

Step 13.2.1

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

$\frac{3 \cdot 3 - 3 \cdot 3^{2} + 6 \cdot 3}{3^{2} + 2 \cdot 3^{2} + 3 \cdot 3}$

Step 13.2.2

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

$\frac{3 \cdot 3 + 3 (- 3^{2}) + 6 \cdot 3}{3^{2} + 2 \cdot 3^{2} + 3 \cdot 3}$

Step 13.2.3

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

$\frac{3 \cdot (3 - 3^{2}) + 6 \cdot 3}{3^{2} + 2 \cdot 3^{2} + 3 \cdot 3}$

Step 13.2.4

Factor $3$ out of $6 \cdot 3$ .

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

Step 13.2.5

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

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

Step 13.2.6

Cancel the common factors.

Tap for more steps...

Step 13.2.6.1

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

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

Step 13.2.6.2

Factor $3$ out of $2 \cdot 3^{2}$ .

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

Step 13.2.6.3

Factor $3$ out of $3 \cdot 3 + 3 (2 \cdot 3)$ .

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

Step 13.2.6.4

Factor $3$ out of $3 \cdot 3$ .

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

Step 13.2.6.5

Factor $3$ out of $3 \cdot (3 + 2 \cdot 3) + 3 (3)$ .

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

Step 13.2.6.6

Cancel the common factor.

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

Step 13.2.6.7

Rewrite the expression.

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

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

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

Step 13.3

Simplify the numerator.

Tap for more steps...

Step 13.3.1

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

$\frac{3 - 1 \cdot 9 + 2 \cdot 3}{3 + 2 \cdot 3 + 3}$

Step 13.3.2

Multiply $- 1$ by $9$ .

$\frac{3 - 9 + 2 \cdot 3}{3 + 2 \cdot 3 + 3}$

Step 13.3.3

Multiply $2$ by $3$ .

$\frac{3 - 9 + 6}{3 + 2 \cdot 3 + 3}$

Step 13.3.4

Subtract $9$ from $3$ .

$\frac{- 6 + 6}{3 + 2 \cdot 3 + 3}$

Step 13.3.5

Add $- 6$ and $6$ .

$\frac{0}{3 + 2 \cdot 3 + 3}$

$\frac{0}{3 + 2 \cdot 3 + 3}$

Step 13.4

Simplify the denominator.

Tap for more steps...

Step 13.4.1

Multiply $2$ by $3$ .

$\frac{0}{3 + 6 + 3}$

Step 13.4.2

Add $3$ and $6$ .

$\frac{0}{9 + 3}$

Step 13.4.3

Add $9$ and $3$ .

$\frac{0}{12}$

$\frac{0}{12}$

Step 13.5

Divide $0$ by $12$ .

$0$

$0$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$