Calculus Examples

$lim_{x \to - 2 π} \sqrt{x + 11} \cos (x + 2 π)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 2 π$ .

$lim_{x \to - 2 π} \sqrt{x + 11} \cdot lim_{x \to - 2 π} \cos (x + 2 π)$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to - 2 π} x + 11} \cdot lim_{x \to - 2 π} \cos (x + 2 π)$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2 π$ .

$\sqrt{lim_{x \to - 2 π} x + lim_{x \to - 2 π} 11} \cdot lim_{x \to - 2 π} \cos (x + 2 π)$

Step 4

Evaluate the limit of $11$ which is constant as $x$ approaches $- 2 π$ .

$\sqrt{lim_{x \to - 2 π} x + 11} \cdot lim_{x \to - 2 π} \cos (x + 2 π)$

Step 5

Move the limit inside the trig function because cosine is continuous.

$\sqrt{lim_{x \to - 2 π} x + 11} \cdot \cos (lim_{x \to - 2 π} x + 2 π)$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2 π$ .

$\sqrt{lim_{x \to - 2 π} x + 11} \cdot \cos (lim_{x \to - 2 π} x + lim_{x \to - 2 π} 2 π)$

Step 7

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

$\sqrt{lim_{x \to - 2 π} x + 11} \cdot \cos (lim_{x \to - 2 π} x + 2 π)$

Step 8

Evaluate the limits by plugging in $- 2 π$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $- 2 π$ for $x$ .

$\sqrt{- 2 π + 11} \cdot \cos (lim_{x \to - 2 π} x + 2 π)$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 2 π$ for $x$ .

$\sqrt{- 2 π + 11} \cdot \cos (- 2 π + 2 π)$

Step 9

Simplify the answer.

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Step 9.1

Add $- 2 π$ and $2 π$ .

$\sqrt{- 2 π + 11} \cdot \cos (0)$

Step 9.2

The exact value of $\cos (0)$ is $1$ .

$\sqrt{- 2 π + 11} \cdot 1$

Step 9.3

Multiply $\sqrt{- 2 π + 11}$ by $1$ .

$\sqrt{- 2 π + 11}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\sqrt{- 2 π + 11}$

Decimal Form:

$2.17182289 \dots$