Calculus Examples

$lim_{x \to 0} e^{x} \cos (x)$

Step 1

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

$lim_{x \to 0} e^{x} \cdot lim_{x \to 0} \cos (x)$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 0} x} \cdot lim_{x \to 0} \cos (x)$

Step 3

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

$e^{lim_{x \to 0} x} \cdot \cos (lim_{x \to 0} x)$

Step 4

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

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Step 4.1

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

$e^{0} \cdot \cos (lim_{x \to 0} x)$

Step 4.2

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

$e^{0} \cdot \cos (0)$

Step 5

Simplify the answer.

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Step 5.1

Anything raised to $0$ is $1$ .

$1 \cdot \cos (0)$

Step 5.2

Multiply $\cos (0)$ by $1$ .

$\cos (0)$

Step 5.3

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

$1$