Calculus Examples

$lim_{x \to 0} x^{2} \cos (20 π x)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} x^{2} \cdot lim_{x \to 0} \cos (20 π x)$

Step 2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} \cos (20 π x)$

Step 3

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

${(lim_{x \to 0} x)}^{2} \cdot \cos (lim_{x \to 0} 20 π x)$

Step 4

Move the term $20 π$ outside of the limit because it is constant with respect to $x$ .

${(lim_{x \to 0} x)}^{2} \cdot \cos (20 π lim_{x \to 0} x)$

Step 5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$0^{2} \cdot \cos (20 π lim_{x \to 0} x)$

Step 5.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$0^{2} \cdot \cos (20 π \cdot 0)$

Step 6

Simplify the answer.

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Step 6.1

Raising $0$ to any positive power yields $0$ .

$0 \cdot \cos (20 π \cdot 0)$

Step 6.2

Multiply $20 π \cdot 0$ .

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Step 6.2.1

Multiply $0$ by $20$ .

$0 \cdot \cos (0 π)$

Step 6.2.2

Multiply $0$ by $π$ .

$0 \cdot \cos (0)$

Step 6.3

The exact value of $\cos (0)$ is $1$ .

$0 \cdot 1$

Step 6.4

Multiply $0$ by $1$ .

$0$