Calculus Examples

$lim_{x \to 0} (x^{2} - 1) (2 - \cos (x))$

Step 1

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

$lim_{x \to 0} x^{2} - 1 \cdot lim_{x \to 0} 2 - \cos (x)$

Step 2

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

$(lim_{x \to 0} x^{2} - lim_{x \to 0} 1) \cdot lim_{x \to 0} 2 - \cos (x)$

Step 3

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

$({(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 1) \cdot lim_{x \to 0} 2 - \cos (x)$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$({(lim_{x \to 0} x)}^{2} - 1 \cdot 1) \cdot lim_{x \to 0} 2 - \cos (x)$

Step 5

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

$({(lim_{x \to 0} x)}^{2} - 1 \cdot 1) \cdot (lim_{x \to 0} 2 - lim_{x \to 0} \cos (x))$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$({(lim_{x \to 0} x)}^{2} - 1 \cdot 1) \cdot (2 - lim_{x \to 0} \cos (x))$

Step 7

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

$({(lim_{x \to 0} x)}^{2} - 1 \cdot 1) \cdot (2 - \cos (lim_{x \to 0} x))$

Step 8

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

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Step 8.1

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

$(0^{2} - 1 \cdot 1) \cdot (2 - \cos (lim_{x \to 0} x))$

Step 8.2

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

$(0^{2} - 1 \cdot 1) \cdot (2 - \cos (0))$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$(0 - 1 \cdot 1) \cdot (2 - \cos (0))$

Step 9.1.2

Multiply $- 1$ by $1$ .

$(0 - 1) \cdot (2 - \cos (0))$

Step 9.2

Subtract $1$ from $0$ .

$- 1 \cdot (2 - \cos (0))$

Step 9.3

Simplify each term.

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Step 9.3.1

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

$- 1 \cdot (2 - 1 \cdot 1)$

Step 9.3.2

Multiply $- 1$ by $1$ .

$- 1 \cdot (2 - 1)$

Step 9.4

Subtract $1$ from $2$ .

$- 1 \cdot 1$

Step 9.5

Multiply $- 1$ by $1$ .

$- 1$