Calculus Examples

$lim_{x \to 8} \frac{\arctan (x)}{x + 2} \cdot (x + \sqrt{x^{2} + 2 x})$

Step 1

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

$lim_{x \to 8} \frac{\arctan (x)}{x + 2} \cdot lim_{x \to 8} x + \sqrt{x^{2} + 2 x}$

Step 2

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

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + 2} \cdot lim_{x \to 8} x + \sqrt{x^{2} + 2 x}$

Step 3

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

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + lim_{x \to 8} 2} \cdot lim_{x \to 8} x + \sqrt{x^{2} + 2 x}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + 2} \cdot lim_{x \to 8} x + \sqrt{x^{2} + 2 x}$

Step 5

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

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + 2} \cdot (lim_{x \to 8} x + lim_{x \to 8} \sqrt{x^{2} + 2 x})$

Step 6

Move the limit under the radical sign.

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + 2} \cdot (lim_{x \to 8} x + \sqrt{lim_{x \to 8} x^{2} + 2 x})$

Step 7

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

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + 2} \cdot (lim_{x \to 8} x + \sqrt{lim_{x \to 8} x^{2} + lim_{x \to 8} 2 x})$

Step 8

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + 2} \cdot (lim_{x \to 8} x + \sqrt{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 2 x})$

Step 9

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to 8} \arctan (x)}{lim_{x \to 8} x + 2} \cdot (lim_{x \to 8} x + \sqrt{{(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x})$

Step 10

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

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Step 10.1

Evaluate the limit of $\arctan (x)$ by plugging in $8$ for $x$ .

$\frac{\arctan (8)}{lim_{x \to 8} x + 2} \cdot (lim_{x \to 8} x + \sqrt{{(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x})$

Step 10.2

Evaluate $\arctan (8)$ .

$\frac{1.44644133}{lim_{x \to 8} x + 2} \cdot (lim_{x \to 8} x + \sqrt{{(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x})$

Step 10.3

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

$\frac{1.44644133}{8 + 2} \cdot (lim_{x \to 8} x + \sqrt{{(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x})$

Step 10.4

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

$\frac{1.44644133}{8 + 2} \cdot (8 + \sqrt{{(lim_{x \to 8} x)}^{2} + 2 lim_{x \to 8} x})$

Step 10.5

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

$\frac{1.44644133}{8 + 2} \cdot (8 + \sqrt{8^{2} + 2 lim_{x \to 8} x})$

Step 10.6

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

$\frac{1.44644133}{8 + 2} \cdot (8 + \sqrt{8^{2} + 2 \cdot 8})$

Step 11

Simplify the answer.

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Step 11.1

Add $8$ and $2$ .

$\frac{1.44644133}{10} \cdot (8 + \sqrt{8^{2} + 2 \cdot 8})$

Step 11.2

Divide $1.44644133$ by $10$ .

$0.14464413 \cdot (8 + \sqrt{8^{2} + 2 \cdot 8})$

Step 11.3

Simplify each term.

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Step 11.3.1

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

$0.14464413 \cdot (8 + \sqrt{64 + 2 \cdot 8})$

Step 11.3.2

Multiply $2$ by $8$ .

$0.14464413 \cdot (8 + \sqrt{64 + 16})$

Step 11.3.3

Add $64$ and $16$ .

$0.14464413 \cdot (8 + \sqrt{80})$

Step 11.3.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 11.3.4.1

Factor $16$ out of $80$ .

$0.14464413 \cdot (8 + \sqrt{16 (5)})$

Step 11.3.4.2

Rewrite $16$ as $4^{2}$ .

$0.14464413 \cdot (8 + \sqrt{4^{2} \cdot 5})$

Step 11.3.5

Pull terms out from under the radical.

$0.14464413 \cdot (8 + 4 \sqrt{5})$

Step 11.4

Multiply $0.14464413$ by $8 + 4 \sqrt{5}$ .

$2.45088952$