Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 1.1.2

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

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

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

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

Step 1.1.3.1.4

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

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

Step 1.1.3.1.5

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.1.2

Subtract $1$ from $0$ .

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

Step 1.1.3.3.1.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.1.3.3.1.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 1.1.3.3.1.3.1.1

Raise $- 1$ to the power of $1$ .

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

Step 1.1.3.3.1.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.3.3.1.3.2

Add $1$ and $2$ .

$\frac{0}{1 + {(- 1)}^{3}}$

Step 1.1.3.3.1.4

Raise $- 1$ to the power of $3$ .

$\frac{0}{1 - 1}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{x}{1 - {(x - 1)}^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [1 - {(x - 1)}^{2}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [1 - {(x - 1)}^{2}]}$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [1 - {(x - 1)}^{2}]}$

Step 1.3.3

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [1 - ((x - 1) (x - 1))]}$

Step 1.3.4

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.3.4.1

Apply the distributive property.

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [1 - (x (x - 1) - 1 (x - 1))]}$

Step 1.3.4.2

Apply the distributive property.

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [1 - (x \cdot x + x \cdot - 1 - 1 (x - 1))]}$

Step 1.3.4.3

Apply the distributive property.

$lim_{x \to 0} \frac{1}{\frac{d}{d x} [1 - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)]}$

Step 1.3.5

Simplify and combine like terms.

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Step 1.3.5.1

Simplify each term.

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Step 1.3.5.1.1

Multiply $x$ by $x$ .

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

Step 1.3.5.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.3.5.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.5.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.5.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.5.2

Subtract $x$ from $- x$ .

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

Step 1.3.6

By the Sum Rule, the derivative of $1 - (x^{2} - 2 x + 1)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- (x^{2} - 2 x + 1)]$ .

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

Step 1.3.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.3.8

Evaluate $\frac{d}{d x} [- (x^{2} - 2 x + 1)]$ .

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Step 1.3.8.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x^{2} - 2 x + 1)$ with respect to $x$ is $- \frac{d}{d x} [x^{2} - 2 x + 1]$ .

$lim_{x \to 0} \frac{1}{0 - \frac{d}{d x} [x^{2} - 2 x + 1]}$

Step 1.3.8.2

By the Sum Rule, the derivative of $x^{2} - 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$ .

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

Step 1.3.8.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{1}{0 - (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1])}$

Step 1.3.8.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{1}{0 - (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [1])}$

Step 1.3.8.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 1.3.8.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.3.8.7

Multiply $- 2$ by $1$ .

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

Step 1.3.8.8

Add $2 x - 2$ and $0$ .

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

Step 1.3.9

Simplify.

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Step 1.3.9.1

Apply the distributive property.

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

Step 1.3.9.2

Combine terms.

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Step 1.3.9.2.1

Multiply $2$ by $- 1$ .

$lim_{x \to 0} \frac{1}{0 - 2 x - - 2}$

Step 1.3.9.2.2

Multiply $- 1$ by $- 2$ .

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

Step 1.3.9.2.3

Subtract $- (- 2 x + 2)$ from $0$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

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

Step 4

Simplify the denominator.

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Step 4.1

Multiply $- 2$ by $0$ .

$\frac{1}{0 + 2}$

Step 4.2

Add $0$ and $2$ .

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$