Calculus Examples

$lim_{x \to 1} \frac{3 - 2 x - x^{2}}{- 4 + 3 x + x^{2}}$

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.

$\frac{lim_{x \to 1} 3 - 2 x - x^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

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 $1$ .

$\frac{lim_{x \to 1} 3 - lim_{x \to 1} 2 x - lim_{x \to 1} x^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.2

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\frac{3 - lim_{x \to 1} 2 x - lim_{x \to 1} x^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.3

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

$\frac{3 - 2 lim_{x \to 1} x - lim_{x \to 1} x^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.4

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

$\frac{3 - 2 lim_{x \to 1} x - {(lim_{x \to 1} x)}^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

$\frac{3 - 2 \cdot 1 - {(lim_{x \to 1} x)}^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.5.2

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

$\frac{3 - 2 \cdot 1 - 1^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

Multiply $- 2$ by $1$ .

$\frac{3 - 2 - 1^{2}}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.6.1.2

One to any power is one.

$\frac{3 - 2 - 1 \cdot 1}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.6.1.3

Multiply $- 1$ by $1$ .

$\frac{3 - 2 - 1}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.6.2

Subtract $2$ from $3$ .

$\frac{1 - 1}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.2.6.3

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} - 4 + 3 x + x^{2}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{- lim_{x \to 1} 4 + lim_{x \to 1} 3 x + lim_{x \to 1} x^{2}}$

Step 1.1.3.2

Evaluate the limit of $4$ which is constant as $x$ approaches $1$ .

$\frac{0}{- 1 \cdot 4 + lim_{x \to 1} 3 x + lim_{x \to 1} x^{2}}$

Step 1.1.3.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{- 4 + 3 lim_{x \to 1} x + lim_{x \to 1} x^{2}}$

Step 1.1.3.4

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

$\frac{0}{- 4 + 3 lim_{x \to 1} x + {(lim_{x \to 1} x)}^{2}}$

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

$\frac{0}{- 4 + 3 \cdot 1 + {(lim_{x \to 1} x)}^{2}}$

Step 1.1.3.5.2

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

$\frac{0}{- 4 + 3 \cdot 1 + 1^{2}}$

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

Multiply $3$ by $1$ .

$\frac{0}{- 4 + 3 + 1^{2}}$

Step 1.1.3.6.1.2

One to any power is one.

$\frac{0}{- 4 + 3 + 1}$

Step 1.1.3.6.2

Add $- 4$ and $3$ .

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

Step 1.1.3.6.3

Add $- 1$ and $1$ .

$\frac{0}{0}$

Step 1.1.3.6.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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.

$lim_{x \to 1} \frac{3 - 2 x - x^{2}}{- 4 + 3 x + x^{2}} = lim_{x \to 1} \frac{\frac{d}{d x} [3 - 2 x - x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [3 - 2 x - x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.2

By the Sum Rule, the derivative of $3 - 2 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- x^{2}]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [3] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{0 + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 1.3.4.1

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

$lim_{x \to 1} \frac{0 - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.4.2

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

$lim_{x \to 1} \frac{0 - 2 \cdot 1 + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.4.3

Multiply $- 2$ by $1$ .

$lim_{x \to 1} \frac{0 - 2 + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.5

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 1.3.5.1

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

$lim_{x \to 1} \frac{0 - 2 - \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.5.2

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

$lim_{x \to 1} \frac{0 - 2 - (2 x)}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.5.3

Multiply $2$ by $- 1$ .

$lim_{x \to 1} \frac{0 - 2 - 2 x}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.6

Simplify.

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Step 1.3.6.1

Subtract $2$ from $0$ .

$lim_{x \to 1} \frac{- 2 - 2 x}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.6.2

Reorder terms.

$lim_{x \to 1} \frac{- 2 x - 2}{\frac{d}{d x} [- 4 + 3 x + x^{2}]}$

Step 1.3.7

By the Sum Rule, the derivative of $- 4 + 3 x + x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 4] + \frac{d}{d x} [3 x] + \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 1} \frac{- 2 x - 2}{\frac{d}{d x} [- 4] + \frac{d}{d x} [3 x] + \frac{d}{d x} [x^{2}]}$

Step 1.3.8

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 1.3.9

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.3.9.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 1.3.9.2

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

$lim_{x \to 1} \frac{- 2 x - 2}{0 + 3 \cdot 1 + \frac{d}{d x} [x^{2}]}$

Step 1.3.9.3

Multiply $3$ by $1$ .

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

Step 1.3.10

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

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

Step 1.3.11

Simplify.

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Step 1.3.11.1

Add $0$ and $3$ .

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

Step 1.3.11.2

Reorder terms.

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

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 $1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

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

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 $x$ .

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

Step 3.2

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

$\frac{- 2 \cdot 1 - 1 \cdot 2}{2 \cdot 1 + 3}$

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 $1$ .

$\frac{- 2 - 1 \cdot 2}{2 \cdot 1 + 3}$

Step 4.1.2

Multiply $- 1$ by $2$ .

$\frac{- 2 - 2}{2 \cdot 1 + 3}$

Step 4.1.3

Subtract $2$ from $- 2$ .

$\frac{- 4}{2 \cdot 1 + 3}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $1$ .

$\frac{- 4}{2 + 3}$

Step 4.2.2

Add $2$ and $3$ .

$\frac{- 4}{5}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{4}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{4}{5}$

Decimal Form:

$- 0.8$