Calculus Examples

$\frac{lim_{x \to 0} 2 (x + h) - 2 x}{h}$

Step 1

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

$\frac{lim_{x \to 0} 2 (x + h) - lim_{x \to 0} 2 x}{h}$

Step 2

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

$\frac{2 lim_{x \to 0} x + h - lim_{x \to 0} 2 x}{h}$

Step 3

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

$\frac{2 (lim_{x \to 0} x + lim_{x \to 0} h) - lim_{x \to 0} 2 x}{h}$

Step 4

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

$\frac{2 (lim_{x \to 0} x + h) - lim_{x \to 0} 2 x}{h}$

Step 5

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

$\frac{2 (lim_{x \to 0} x + h) - 2 lim_{x \to 0} x}{h}$

Step 6

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

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Step 6.1

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

$\frac{2 (0 + h) - 2 lim_{x \to 0} x}{h}$

Step 6.2

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

$\frac{2 (0 + h) - 2 \cdot 0}{h}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor $2$ out of $2 (0 + h) - 2 \cdot 0$ .

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Step 7.1.1.1

Factor $2$ out of $- 2 \cdot 0$ .

$\frac{2 (0 + h) + 2 (- 0)}{h}$

Step 7.1.1.2

Factor $2$ out of $2 (0 + h) + 2 (- 0)$ .

$\frac{2 (0 + h - 0)}{h}$

Step 7.1.2

Multiply $- 1$ by $0$ .

$\frac{2 (0 + h + 0)}{h}$

Step 7.1.3

Add $0 + h$ and $0$ .

$\frac{2 (0 + h)}{h}$

Step 7.1.4

Add $0$ and $h$ .

$\frac{2 h}{h}$

Step 7.2

Cancel the common factor of $h$ .

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Step 7.2.1

Cancel the common factor.

$\frac{2 h}{h}$

Step 7.2.2

Divide $2$ by $1$ .

$2$