Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

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

$8 \cdot 0.99219766$

Step 6.2

Multiply $8$ by $0.99219766$ .

$7.93758133$