Calculus Examples

$lim_{x \to 0} (x - 5) \cos (x)$

Step 1

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

$lim_{x \to 0} x - 5 \cdot lim_{x \to 0} \cos (x)$

Step 2

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

$(lim_{x \to 0} x - lim_{x \to 0} 5) \cdot lim_{x \to 0} \cos (x)$

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$(lim_{x \to 0} x - 1 \cdot 5) \cdot lim_{x \to 0} \cos (x)$

Step 4

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

$(lim_{x \to 0} x - 1 \cdot 5) \cdot \cos (lim_{x \to 0} x)$

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 $x$ .

$(0 - 1 \cdot 5) \cdot \cos (lim_{x \to 0} x)$

Step 5.2

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

$(0 - 1 \cdot 5) \cdot \cos (0)$

Step 6

Simplify the answer.

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Step 6.1

Multiply $- 1$ by $5$ .

$(0 - 5) \cdot \cos (0)$

Step 6.2

Subtract $5$ from $0$ .

$- 5 \cdot \cos (0)$

Step 6.3

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

$- 5 \cdot 1$

Step 6.4

Multiply $- 5$ by $1$ .

$- 5$