Calculus Examples

$lim_{x \to 8} \arctan (1 + \sqrt{x})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$1 + \sqrt{4 (2)}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$1 + \sqrt{2^{2} \cdot 2}$

Step 1.2

Pull terms out from under the radical.

$1 + 2 \sqrt{2}$

Step 2

Substitute $t$ for $1 + \sqrt{x}$ and let $t$ approach $1 + 2 \sqrt{2}$ since $lim_{x \to 8} 1 + \sqrt{x} = 1 + 2 \sqrt{2}$ .

$lim_{t \to 1 + 2 \sqrt{2}} \arctan (t)$

Step 3

Evaluate the limits by plugging in $1 + 2 \sqrt{2}$ for all occurrences of $t$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.1.1

Factor $4$ out of $8$ .

$\arctan (1 + \sqrt{4 (2)})$

Step 3.1.1.2

Rewrite $4$ as $2^{2}$ .

$\arctan (1 + \sqrt{2^{2} \cdot 2})$

Step 3.1.2

Pull terms out from under the radical.

$\arctan (1 + 2 \sqrt{2})$

Step 3.2

Evaluate $\arctan (1 + 2 \sqrt{2})$ .

$1.31530095$