Calculus Examples

$lim_{n \to 8} \sqrt{20 + a (n)}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt{lim_{n \to 8} 20 + a (n)}$

Step 1.2

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

$\sqrt{lim_{n \to 8} 20 + lim_{n \to 8} a (n)}$

Step 1.3

Evaluate the limit of $20$ which is constant as $n$ approaches $8$ .

$\sqrt{20 + lim_{n \to 8} a (n)}$

Step 1.4

Move the term $a$ outside of the limit because it is constant with respect to $n$ .

$\sqrt{20 + a lim_{n \to 8} n}$

Step 2

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

$\sqrt{20 + a \cdot 8}$

Step 3

Simplify the answer.

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Step 3.1

Factor $4$ out of $20 + a \cdot 8$ .

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Step 3.1.1

Factor $4$ out of $20$ .

$\sqrt{4 (5) + a \cdot 8}$

Step 3.1.2

Factor $4$ out of $a \cdot 8$ .

$\sqrt{4 (5) + 4 (a \cdot 2)}$

Step 3.1.3

Factor $4$ out of $4 (5) + 4 (a \cdot 2)$ .

$\sqrt{4 (5 + a \cdot 2)}$

Step 3.2

Move $2$ to the left of $a$ .

$\sqrt{4 (5 + 2 \cdot a)}$

Step 3.3

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

$\sqrt{2^{2} (5 + 2 a)}$

Step 3.4

Pull terms out from under the radical.

$2 \sqrt{5 + 2 a}$