Calculus Examples

$lim_{x \to 8} x (1 - \cos (\frac{1}{x}))$

Step 1

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

$lim_{x \to 8} x \cdot lim_{x \to 8} 1 - \cos (\frac{1}{x})$

Step 2

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

$lim_{x \to 8} x \cdot (lim_{x \to 8} 1 - lim_{x \to 8} \cos (\frac{1}{x}))$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot (1 - lim_{x \to 8} \cos (\frac{1}{x}))$

Step 4

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

$lim_{x \to 8} x \cdot (1 - \cos (lim_{x \to 8} \frac{1}{x}))$

Step 5

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

$lim_{x \to 8} x \cdot (1 - \cos (\frac{lim_{x \to 8} 1}{lim_{x \to 8} x}))$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot (1 - \cos (\frac{1}{lim_{x \to 8} x}))$

Step 7

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

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Step 7.1

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

$8 \cdot (1 - \cos (\frac{1}{lim_{x \to 8} x}))$

Step 7.2

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

$8 \cdot (1 - \cos (\frac{1}{8}))$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Evaluate $\cos (\frac{1}{8})$ .

$8 \cdot (1 - 1 \cdot 0.99219766)$

Step 8.1.2

Multiply $- 1$ by $0.99219766$ .

$8 \cdot (1 - 0.99219766)$

Step 8.2

Subtract $0.99219766$ from $1$ .

$8 \cdot 0.00780233$

Step 8.3

Multiply $8$ by $0.00780233$ .

$0.06241866$