Calculus Examples

$lim_{x \to 3} 10 \arctan (x - 2)$

Step 1

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$10 lim_{x \to 3} \arctan (x - 2)$

Step 2

Simplify the answer.

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Step 2.1

Multiply $- 1$ by $2$ .

$10 (3 - 2)$

Step 2.2

Subtract $2$ from $3$ .

$10 \cdot 1$

Step 3

Substitute $t$ for $x - 2$ and let $t$ approach $1$ since $lim_{x \to 3} x - 2 = 1$ .

$10 lim_{t \to 1} \arctan (t)$

Step 4

Evaluate the limits by plugging in $1$ for all occurrences of $t$ .

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Step 4.1

Simplify the answer.

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Step 4.1.1

Multiply $- 1$ by $2$ .

$10 \arctan (3 - 2)$

Step 4.1.2

Subtract $2$ from $3$ .

$10 \arctan (1)$

Step 4.2

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$10 \frac{π}{4}$

Step 5

Simplify the answer.

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Step 5.1

Cancel the common factor of $2$ .

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Step 5.1.1

Factor $2$ out of $10$ .

$2 (5) \frac{π}{4}$

Step 5.1.2

Factor $2$ out of $4$ .

$2 \cdot 5 \frac{π}{2 \cdot 2}$

Step 5.1.3

Cancel the common factor.

$2 \cdot 5 \frac{π}{2 \cdot 2}$

Step 5.1.4

Rewrite the expression.

$5 \frac{π}{2}$

Step 5.2

Combine $5$ and $\frac{π}{2}$ .

$\frac{5 π}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5 π}{2}$

Decimal Form:

$7.85398163 \dots$