Calculus Examples

$lim_{x \to - 1} \frac{x^{2} + x}{x^{3} + 1}$

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.

$\frac{lim_{x \to - 1} x^{2} + x}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{lim_{x \to - 1} x^{2} + lim_{x \to - 1} x}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.2

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

$\frac{{(lim_{x \to - 1} x)}^{2} + lim_{x \to - 1} x}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.3

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 1.1.2.3.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{{(- 1)}^{2} + lim_{x \to - 1} x}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.3.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{{(- 1)}^{2} - 1}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.4

Simplify the answer.

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Step 1.1.2.4.1

Raise $- 1$ to the power of $2$ .

$\frac{1 - 1}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.4.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to - 1} x^{3} + 1}$

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 $- 1$ .

$\frac{0}{lim_{x \to - 1} x^{3} + lim_{x \to - 1} 1}$

Step 1.1.3.1.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

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

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

Step 1.1.3.3.2

Add $- 1$ and $1$ .

$\frac{0}{0}$

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.

$lim_{x \to - 1} \frac{x^{2} + x}{x^{3} + 1} = lim_{x \to - 1} \frac{\frac{d}{d x} [x^{2} + x]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 1} \frac{\frac{d}{d x} [x^{2} + x]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.2

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

$lim_{x \to - 1} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.3

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

$lim_{x \to - 1} \frac{2 x + \frac{d}{d x} [x]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.4

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

$lim_{x \to - 1} \frac{2 x + 1}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.5

By the Sum Rule, the derivative of $x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]$ .

$lim_{x \to - 1} \frac{2 x + 1}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]}$

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to - 1} \frac{2 x + 1}{3 x^{2} + \frac{d}{d x} [1]}$

Step 1.3.7

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

$lim_{x \to - 1} \frac{2 x + 1}{3 x^{2} + 0}$

Step 1.3.8

Add $3 x^{2}$ and $0$ .

$lim_{x \to - 1} \frac{2 x + 1}{3 x^{2}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to - 1} \frac{2 x + 1}{x^{2}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 1$ .

$\frac{1}{3} \cdot \frac{lim_{x \to - 1} 2 x + 1}{lim_{x \to - 1} x^{2}}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{1}{3} \cdot \frac{lim_{x \to - 1} 2 x + lim_{x \to - 1} 1}{lim_{x \to - 1} x^{2}}$

Step 2.4

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

$\frac{1}{3} \cdot \frac{2 lim_{x \to - 1} x + lim_{x \to - 1} 1}{lim_{x \to - 1} x^{2}}$

Step 2.5

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

$\frac{1}{3} \cdot \frac{2 lim_{x \to - 1} x + 1}{lim_{x \to - 1} x^{2}}$

Step 2.6

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

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

Step 3

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 3.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{1}{3} \cdot \frac{2 \cdot - 1 + 1}{{(- 1)}^{2}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $2$ by $- 1$ .

$\frac{1}{3} \cdot \frac{- 2 + 1}{{(- 1)}^{2}}$

Step 4.1.2

Add $- 2$ and $1$ .

$\frac{1}{3} \cdot \frac{- 1}{{(- 1)}^{2}}$

Step 4.2

Raise $- 1$ to the power of $2$ .

$\frac{1}{3} \cdot \frac{- 1}{1}$

Step 4.3

Divide $- 1$ by $1$ .

$\frac{1}{3} \cdot - 1$

Step 4.4

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{- 1}{3}$

Step 4.5

Move the negative in front of the fraction.

$- \frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$