Calculus Examples

$\frac{lim_{x \to 1} {(6 x - 3)}^{2} - 9}{3 x - 3}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} {(6 x - 3)}^{2} - lim_{x \to 1} 9}{3 x - 3}$

Step 1.2

Move the exponent $2$ from ${(6 x - 3)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} 6 x - 3)}^{2} - lim_{x \to 1} 9}{3 x - 3}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} 6 x - lim_{x \to 1} 3)}^{2} - lim_{x \to 1} 9}{3 x - 3}$

Step 1.4

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

$\frac{{(6 lim_{x \to 1} x - lim_{x \to 1} 3)}^{2} - lim_{x \to 1} 9}{3 x - 3}$

Step 1.5

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\frac{{(6 lim_{x \to 1} x - 1 \cdot 3)}^{2} - lim_{x \to 1} 9}{3 x - 3}$

Step 1.6

Evaluate the limit of $9$ which is constant as $x$ approaches $1$ .

$\frac{{(6 lim_{x \to 1} x - 1 \cdot 3)}^{2} - 1 \cdot 9}{3 x - 3}$

Step 2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{{(6 \cdot 1 - 1 \cdot 3)}^{2} - 1 \cdot 9}{3 x - 3}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Multiply $6$ by $1$ .

$\frac{{(6 - 1 \cdot 3)}^{2} - 1 \cdot 9}{3 x - 3}$

Step 3.1.1.2

Multiply $- 1$ by $3$ .

$\frac{{(6 - 3)}^{2} - 1 \cdot 9}{3 x - 3}$

Step 3.1.2

Subtract $3$ from $6$ .

$\frac{3^{2} - 1 \cdot 9}{3 x - 3}$

Step 3.1.3

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

$\frac{9 - 1 \cdot 9}{3 x - 3}$

Step 3.1.4

Multiply $- 1$ by $9$ .

$\frac{9 - 9}{3 x - 3}$

Step 3.1.5

Subtract $9$ from $9$ .

$\frac{0}{3 x - 3}$

Step 3.2

Factor $3$ out of $3 x - 3$ .

Tap for more steps...

Step 3.2.1

Factor $3$ out of $3 x$ .

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

Step 3.2.2

Factor $3$ out of $- 3$ .

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

Step 3.2.3

Factor $3$ out of $3 (x) + 3 (- 1)$ .

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

Step 3.3

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 3.3.1

Factor $3$ out of $0$ .

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

Step 3.3.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

$\frac{0}{x - 1}$

Step 3.4

Divide $0$ by $x - 1$ .

$0$