Calculus Examples

$lim_{x \to 8} \arctan (\frac{2 x^{2} + 2}{9 - 9 x})$

Step 1

Simplify the answer.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Raise $8$ to the power of $2$ .

$\frac{2 \cdot 64 + 2}{9 - 9 \cdot 8}$

Step 1.1.2

Multiply $2$ by $64$ .

$\frac{128 + 2}{9 - 9 \cdot 8}$

Step 1.1.3

Add $128$ and $2$ .

$\frac{130}{9 - 9 \cdot 8}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Multiply $- 9$ by $8$ .

$\frac{130}{9 - 72}$

Step 1.2.2

Subtract $72$ from $9$ .

$\frac{130}{- 63}$

Step 1.3

Move the negative in front of the fraction.

$- \frac{130}{63}$

Step 2

Substitute $t$ for $\frac{2 x^{2} + 2}{9 - 9 x}$ and let $t$ approach $- \frac{130}{63}$ since $lim_{x \to 8} \frac{2 x^{2} + 2}{9 - 9 x} = - \frac{130}{63}$ .

$lim_{t \to - \frac{130}{63}} \arctan (t)$

Step 3

Evaluate the limits by plugging in $- \frac{130}{63}$ for all occurrences of $t$ .

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Step 3.1

Simplify the answer.

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Step 3.1.1

Simplify the numerator.

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Step 3.1.1.1

Raise $8$ to the power of $2$ .

$\arctan (\frac{2 \cdot 64 + 2}{9 - 9 \cdot 8})$

Step 3.1.1.2

Multiply $2$ by $64$ .

$\arctan (\frac{128 + 2}{9 - 9 \cdot 8})$

Step 3.1.1.3

Add $128$ and $2$ .

$\arctan (\frac{130}{9 - 9 \cdot 8})$

Step 3.1.2

Simplify the denominator.

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Step 3.1.2.1

Multiply $- 9$ by $8$ .

$\arctan (\frac{130}{9 - 72})$

Step 3.1.2.2

Subtract $72$ from $9$ .

$\arctan (\frac{130}{- 63})$

Step 3.1.3

Move the negative in front of the fraction.

$\arctan (- \frac{130}{63})$

Step 3.2

Evaluate $\arctan (- \frac{130}{63})$ .

$- 1.11953198$