Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

$8 - 8 \cdot 0.99219766$

Step 7.1.2

Multiply $- 8$ by $0.99219766$ .

$8 - 7.93758133$

Step 7.2

Subtract $7.93758133$ from $8$ .

$0.06241866$