Calculus Examples

$lim_{x \to 8} \frac{\cos (2 x)}{x} + 2$

Step 1

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

$lim_{x \to 8} \frac{\cos (2 x)}{x} + lim_{x \to 8} 2$

Step 2

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

$\frac{lim_{x \to 8} \cos (2 x)}{lim_{x \to 8} x} + lim_{x \to 8} 2$

Step 3

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

$\frac{\cos (lim_{x \to 8} 2 x)}{lim_{x \to 8} x} + lim_{x \to 8} 2$

Step 4

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

$\frac{\cos (2 lim_{x \to 8} x)}{lim_{x \to 8} x} + lim_{x \to 8} 2$

Step 5

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

$\frac{\cos (2 lim_{x \to 8} x)}{lim_{x \to 8} x} + 2$

Step 6

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

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Step 6.1

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

$\frac{\cos (2 \cdot 8)}{lim_{x \to 8} x} + 2$

Step 6.2

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

$\frac{\cos (2 \cdot 8)}{8} + 2$

Step 7

Simplify the answer.

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Step 7.1

Multiply $2$ by $8$ .

$\frac{\cos (16)}{8} + 2$

Step 7.2

Simplify each term.

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Step 7.2.1

Evaluate $\cos (16)$ .

$\frac{0.96126169}{8} + 2$

Step 7.2.2

Divide $0.96126169$ by $8$ .

$0.12015771 + 2$

Step 7.3

Add $0.12015771$ and $2$ .

$2.12015771$