Calculus Examples

$lim_{x \to 8} x (\arctan (\frac{x}{x + 1}) - \frac{π}{4})$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot lim_{x \to 8} \arctan (\frac{x}{x + 1}) - \frac{π}{4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot (lim_{x \to 8} \arctan (\frac{x}{x + 1}) - lim_{x \to 8} \frac{π}{4})$

Step 3

Add $8$ and $1$ .

$lim_{x \to 8} x \cdot (\frac{8}{9} - lim_{x \to 8} \frac{π}{4})$

Step 4

Substitute $t$ for $\frac{x}{x + 1}$ and let $t$ approach $\frac{8}{9}$ since $lim_{x \to 8} \frac{x}{x + 1} = \frac{8}{9}$ .

$lim_{x \to 8} x \cdot (lim_{t \to \frac{8}{9}} \arctan (t) - lim_{x \to 8} \frac{π}{4})$

Step 5

Evaluate the limit of $\frac{π}{4}$ which is constant as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot (lim_{t \to \frac{8}{9}} \arctan (t) - \frac{π}{4})$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$8 \cdot (lim_{t \to \frac{8}{9}} \arctan (t) - \frac{π}{4})$

Step 6.2

Add $8$ and $1$ .

$8 \cdot (\arctan (\frac{8}{9}) - \frac{π}{4})$

Step 6.3

Evaluate $\arctan (\frac{8}{9})$ .

$8 \cdot (0.72664234 - \frac{π}{4})$

Step 7

Simplify the answer.

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Step 7.1

Subtract $\frac{π}{4}$ from $0.72664234$ .

$8 \cdot - 0.05875582$

Step 7.2

Multiply $8$ by $- 0.05875582$ .

$- 0.47004658$