Calculus Examples

$lim_{x \to - 8} x^{4} \cos (\frac{2}{x})$

Step 1

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

$lim_{x \to - 8} x^{4} \cdot lim_{x \to - 8} \cos (\frac{2}{x})$

Step 2

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

${(lim_{x \to - 8} x)}^{4} \cdot lim_{x \to - 8} \cos (\frac{2}{x})$

Step 3

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

${(lim_{x \to - 8} x)}^{4} \cdot \cos (lim_{x \to - 8} \frac{2}{x})$

Step 4

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

${(lim_{x \to - 8} x)}^{4} \cdot \cos (2 lim_{x \to - 8} \frac{1}{x})$

Step 5

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

${(lim_{x \to - 8} x)}^{4} \cdot \cos (2 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} x})$

Step 6

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

${(lim_{x \to - 8} x)}^{4} \cdot \cos (2 \frac{1}{lim_{x \to - 8} x})$

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 $x$ .

${(- 8)}^{4} \cdot \cos (2 \frac{1}{lim_{x \to - 8} x})$

Step 7.2

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

${(- 8)}^{4} \cdot \cos (2 (\frac{1}{- 8}))$

Step 8

Simplify the answer.

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Step 8.1

Raise $- 8$ to the power of $4$ .

$4096 \cdot \cos (2 (\frac{1}{- 8}))$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Factor $2$ out of $- 8$ .

$4096 \cdot \cos (2 \frac{1}{2 (- 4)})$

Step 8.2.2

Cancel the common factor.

$4096 \cdot \cos (2 \frac{1}{2 \cdot - 4})$

Step 8.2.3

Rewrite the expression.

$4096 \cdot \cos (\frac{1}{- 4})$

Step 8.3

Move the negative in front of the fraction.

$4096 \cdot \cos (- \frac{1}{4})$

Step 8.4

Evaluate $\cos (- \frac{1}{4})$ .

$4096 \cdot 0.96891242$

Step 8.5

Multiply $4096$ by $0.96891242$ .

$3968.66527932$