Calculus Examples

$\frac{lim_{x \to 0} {(1 + x)}^{2} - 1}{x}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 0} {(1 + x)}^{2} - lim_{x \to 0} 1}{x}$

Step 1.2

Move the exponent $2$ from ${(1 + x)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} 1 + x)}^{2} - lim_{x \to 0} 1}{x}$

Step 1.3

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

$\frac{{(lim_{x \to 0} 1 + lim_{x \to 0} x)}^{2} - lim_{x \to 0} 1}{x}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{{(1 + lim_{x \to 0} x)}^{2} - lim_{x \to 0} 1}{x}$

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 2

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

$\frac{{(1 + 0)}^{2} - 1 \cdot 1}{x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $1$ and $0$ .

$\frac{1^{2} - 1 \cdot 1}{x}$

Step 3.1.2

One to any power is one.

$\frac{1 - 1 \cdot 1}{x}$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{x}$

Step 3.1.4

Subtract $1$ from $1$ .

$\frac{0}{x}$

Step 3.2

Divide $0$ by $x$ .

$0$