Calculus Examples

$lim_{x \to 0} \frac{\cos (x) + 3 e^{x}}{2 e^{x}}$

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \frac{\cos (x) + 3 e^{x}}{e^{x}}$

Step 2

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

$\frac{1}{2} \cdot \frac{lim_{x \to 0} \cos (x) + 3 e^{x}}{lim_{x \to 0} e^{x}}$

Step 3

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

$\frac{1}{2} \cdot \frac{lim_{x \to 0} \cos (x) + lim_{x \to 0} 3 e^{x}}{lim_{x \to 0} e^{x}}$

Step 4

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

$\frac{1}{2} \cdot \frac{\cos (lim_{x \to 0} x) + lim_{x \to 0} 3 e^{x}}{lim_{x \to 0} e^{x}}$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{\cos (lim_{x \to 0} x) + 3 lim_{x \to 0} e^{x}}{lim_{x \to 0} e^{x}}$

Step 6

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{\cos (lim_{x \to 0} x) + 3 e^{lim_{x \to 0} x}}{lim_{x \to 0} e^{x}}$

Step 7

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{\cos (lim_{x \to 0} x) + 3 e^{lim_{x \to 0} x}}{e^{lim_{x \to 0} x}}$

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 $x$ .

$\frac{1}{2} \cdot \frac{\cos (0) + 3 e^{lim_{x \to 0} x}}{e^{lim_{x \to 0} x}}$

Step 8.2

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

$\frac{1}{2} \cdot \frac{\cos (0) + 3 e^{0}}{e^{lim_{x \to 0} x}}$

Step 8.3

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

$\frac{1}{2} \cdot \frac{\cos (0) + 3 e^{0}}{e^{0}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{1}{2} \cdot \frac{1 + 3 e^{0}}{e^{0}}$

Step 9.1.2

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{1 + 3 \cdot 1}{e^{0}}$

Step 9.1.3

Multiply $3$ by $1$ .

$\frac{1}{2} \cdot \frac{1 + 3}{e^{0}}$

Step 9.1.4

Add $1$ and $3$ .

$\frac{1}{2} \cdot \frac{4}{e^{0}}$

Step 9.2

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{4}{1}$

Step 9.3

Cancel the common factor of $2$ .

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Step 9.3.1

Factor $2$ out of $4$ .

$\frac{1}{2} \cdot \frac{2 (2)}{1}$

Step 9.3.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot 2}{1}$

Step 9.3.3

Rewrite the expression.

$2$