Calculus Examples

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

Step 1

Simplify the answer.

Tap for more steps...

Step 1.1

Cancel the common factor of $3$ and $6 \cdot 3 - 1 \cdot 9$ .

Tap for more steps...

Step 1.1.1

Factor $- 1$ out of $6 \cdot 3$ .

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

Step 1.1.2

Factor $- 1$ out of $- 1 \cdot 9$ .

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

Step 1.1.3

Factor $- 1$ out of $- (- 6 \cdot 3) - 1 (9)$ .

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

Step 1.1.4

Rewrite $- (- 6 \cdot 3 + 9)$ as $- 1 (- 6 \cdot 3 + 9)$ .

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

Step 1.1.5

Factor $3$ out of $3$ .

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

Step 1.1.6

Cancel the common factors.

Tap for more steps...

Step 1.1.6.1

Factor $3$ out of $- 1 (- 6 \cdot 3 + 9)$ .

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

Step 1.1.6.2

Cancel the common factor.

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

Step 1.1.6.3

Rewrite the expression.

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

Step 1.2

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 1.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 1.2.2

Move the negative in front of the fraction.

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

Step 1.3

Simplify the denominator.

Tap for more steps...

Step 1.3.1

Multiply $- 2$ by $3$ .

$2 (- \frac{1}{- 6 + 3})$

Step 1.3.2

Add $- 6$ and $3$ .

$2 (- \frac{1}{- 3})$

Step 1.4

Move the negative in front of the fraction.

$2 (- - \frac{1}{3})$

Step 1.5

Multiply $- - \frac{1}{3}$ .

Tap for more steps...

Step 1.5.1

Multiply $- 1$ by $- 1$ .

$2 (1 (\frac{1}{3}))$

Step 1.5.2

Multiply $\frac{1}{3}$ by $1$ .

$2 (\frac{1}{3})$

Step 1.6

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3}$

Step 2

Substitute $t$ for $\frac{x^{2} - 9}{3 x^{2} - 9 x}$ and let $t$ approach $\frac{2}{3}$ since $lim_{x \to 3} \frac{x^{2} - 9}{3 x^{2} - 9 x} = \frac{2}{3}$ .

$lim_{t \to \frac{2}{3}} \arctan (t)$

Step 3

Evaluate the limits by plugging in $\frac{2}{3}$ for all occurrences of $t$ .

Tap for more steps...

Step 3.1

Simplify the answer.

Tap for more steps...

Step 3.1.1

Cancel the common factor of $3$ and $6 \cdot 3 - 1 \cdot 9$ .

Tap for more steps...

Step 3.1.1.1

Factor $- 1$ out of $6 \cdot 3$ .

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

Step 3.1.1.2

Factor $- 1$ out of $- 1 \cdot 9$ .

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

Step 3.1.1.3

Factor $- 1$ out of $- (- 6 \cdot 3) - 1 (9)$ .

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

Step 3.1.1.4

Rewrite $- (- 6 \cdot 3 + 9)$ as $- 1 (- 6 \cdot 3 + 9)$ .

$\arctan (2 \frac{3}{- 1 (- 6 \cdot 3 + 9)})$

Step 3.1.1.5

Factor $3$ out of $3$ .

$\arctan (2 \frac{3 (1)}{- 1 (- 6 \cdot 3 + 9)})$

Step 3.1.1.6

Cancel the common factors.

Tap for more steps...

Step 3.1.1.6.1

Factor $3$ out of $- 1 (- 6 \cdot 3 + 9)$ .

$\arctan (2 \frac{3 (1)}{3 (- 1 (- 2 \cdot 3 + 3))})$

Step 3.1.1.6.2

Cancel the common factor.

$\arctan (2 \frac{3 \cdot 1}{3 (- 1 (- 2 \cdot 3 + 3))})$

Step 3.1.1.6.3

Rewrite the expression.

$\arctan (2 \frac{1}{- 1 (- 2 \cdot 3 + 3)})$

Step 3.1.2

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 3.1.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$\arctan (2 \frac{- 1 (- 1)}{- 1 (- 2 \cdot 3 + 3)})$

Step 3.1.2.2

Move the negative in front of the fraction.

$\arctan (2 (- \frac{1}{- 2 \cdot 3 + 3}))$

Step 3.1.3

Simplify the denominator.

Tap for more steps...

Step 3.1.3.1

Multiply $- 2$ by $3$ .

$\arctan (2 (- \frac{1}{- 6 + 3}))$

Step 3.1.3.2

Add $- 6$ and $3$ .

$\arctan (2 (- \frac{1}{- 3}))$

Step 3.1.4

Move the negative in front of the fraction.

$\arctan (2 (- - \frac{1}{3}))$

Step 3.1.5

Multiply $- - \frac{1}{3}$ .

Tap for more steps...

Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

$\arctan (2 (1 (\frac{1}{3})))$

Step 3.1.5.2

Multiply $\frac{1}{3}$ by $1$ .

$\arctan (2 (\frac{1}{3}))$

Step 3.1.6

Combine $2$ and $\frac{1}{3}$ .

$\arctan (\frac{2}{3})$

Step 3.2

Evaluate $\arctan (\frac{2}{3})$ .

$0.5880026$