Calculus Examples

$\frac{lim_{x \to 0} 2 e^{x} - 2 + x}{5 x}$

Step 1

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

$\frac{lim_{x \to 0} 2 e^{x} - lim_{x \to 0} 2 + lim_{x \to 0} x}{5 x}$

Step 2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 0} e^{x} - lim_{x \to 0} 2 + lim_{x \to 0} x}{5 x}$

Step 3

Move the limit into the exponent.

$\frac{2 e^{lim_{x \to 0} x} - lim_{x \to 0} 2 + lim_{x \to 0} x}{5 x}$

Step 4

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

$\frac{2 e^{lim_{x \to 0} x} - 1 \cdot 2 + lim_{x \to 0} x}{5 x}$

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 $x$ .

$\frac{2 e^{0} - 1 \cdot 2 + lim_{x \to 0} x}{5 x}$

Step 5.2

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

$\frac{2 e^{0} - 1 \cdot 2 + 0}{5 x}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Anything raised to $0$ is $1$ .

$\frac{2 \cdot 1 - 1 \cdot 2 + 0}{5 x}$

Step 6.1.2

Multiply $2$ by $1$ .

$\frac{2 - 1 \cdot 2 + 0}{5 x}$

Step 6.1.3

Multiply $- 1$ by $2$ .

$\frac{2 - 2 + 0}{5 x}$

Step 6.1.4

Add $2 - 2$ and $0$ .

$\frac{2 - 2}{5 x}$

Step 6.1.5

Subtract $2$ from $2$ .

$\frac{0}{5 x}$

Step 6.2

Cancel the common factor of $0$ and $5$ .

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Step 6.2.1

Factor $5$ out of $0$ .

$\frac{5 (0)}{5 x}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $5$ out of $5 x$ .

$\frac{5 (0)}{5 (x)}$

Step 6.2.2.2

Cancel the common factor.

$\frac{5 \cdot 0}{5 x}$

Step 6.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 6.3

Divide $0$ by $x$ .

$0$