Calculus Examples

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

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.

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

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 $- \frac{1}{2}$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $- \frac{1}{2}$ .

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

Step 1.1.2.5

Evaluate the limits by plugging in $- \frac{1}{2}$ for all occurrences of $x$ .

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Step 1.1.2.5.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 1.1.2.5.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.2.6.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 1.1.2.6.1.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.1.2.6.1.2

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

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

Step 1.1.2.6.1.3

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

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

Step 1.1.2.6.1.4

Cancel the common factor of $2$ .

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Step 1.1.2.6.1.4.1

Factor $2$ out of $2^{2}$ .

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

Step 1.1.2.6.1.4.2

Cancel the common factor.

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

Step 1.1.2.6.1.4.3

Rewrite the expression.

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

Step 1.1.2.6.1.5

One to any power is one.

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

Step 1.1.2.6.1.6

Multiply $- - \frac{1}{2}$ .

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Step 1.1.2.6.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.2.6.1.6.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 1.1.2.6.1.7

Multiply $- 1$ by $1$ .

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

Step 1.1.2.6.2

Combine the numerators over the common denominator.

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

Step 1.1.2.6.3

Add $1$ and $1$ .

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

Step 1.1.2.6.4

Divide $2$ by $2$ .

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

Step 1.1.2.6.5

Add $- 1$ and $1$ .

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

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 $- \frac{1}{2}$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Evaluate the limit of $1$ which is constant as $x$ approaches $- \frac{1}{2}$ .

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

Step 1.1.3.6

Evaluate the limits by plugging in $- \frac{1}{2}$ for all occurrences of $x$ .

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Step 1.1.3.6.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 1.1.3.6.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 1.1.3.7

Simplify the answer.

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Step 1.1.3.7.1

Simplify each term.

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Step 1.1.3.7.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.3.7.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 1.1.3.7.1.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.1.3.7.1.2

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

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

Step 1.1.3.7.1.3

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

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

Step 1.1.3.7.1.4

Cancel the common factor of $2$ .

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Step 1.1.3.7.1.4.1

Factor $2$ out of $2^{2}$ .

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

Step 1.1.3.7.1.4.2

Cancel the common factor.

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

Step 1.1.3.7.1.4.3

Rewrite the expression.

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

Step 1.1.3.7.1.5

One to any power is one.

$\frac{0}{\frac{1}{2} + 3 (- \frac{1}{2}) + 1}$

Step 1.1.3.7.1.6

Multiply $3 (- \frac{1}{2})$ .

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Step 1.1.3.7.1.6.1

Multiply $- 1$ by $3$ .

$\frac{0}{\frac{1}{2} - 3 (\frac{1}{2}) + 1}$

Step 1.1.3.7.1.6.2

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 1.1.3.7.1.7

Move the negative in front of the fraction.

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

Step 1.1.3.7.2

Combine the numerators over the common denominator.

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

Step 1.1.3.7.3

Subtract $3$ from $1$ .

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

Step 1.1.3.7.4

Divide $- 2$ by $2$ .

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

Step 1.1.3.7.5

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.7.6

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

Undefined

$\frac{0}{0}$

Step 1.1.3.8

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

Multiply $2$ by $2$ .

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

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 $n = 1$ .

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

Step 1.3.4.3

Multiply $- 1$ by $1$ .

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

Step 1.3.5

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

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

Step 1.3.6

Add $4 x - 1$ and $0$ .

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

Step 1.3.7

By the Sum Rule, the derivative of $2 x^{2} + 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

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

Step 1.3.8

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

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Step 1.3.8.1

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

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

Step 1.3.8.2

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

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

Step 1.3.8.3

Multiply $2$ by $2$ .

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

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

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

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 $n = 1$ .

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

Step 1.3.9.3

Multiply $3$ by $1$ .

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

Step 1.3.10

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

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

Step 1.3.11

Add $4 x + 3$ and $0$ .

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

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 $- \frac{1}{2}$ .

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

Step 2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{2}$ .

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

Step 2.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $- \frac{1}{2}$ .

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

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{2}$ .

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

Step 2.6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 2.7

Evaluate the limit of $3$ which is constant as $x$ approaches $- \frac{1}{2}$ .

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

Step 3

Evaluate the limits by plugging in $- \frac{1}{2}$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 3.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 4.1.1.2

Factor $2$ out of $4$ .

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

Step 4.1.1.3

Cancel the common factor.

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

Step 4.1.1.4

Rewrite the expression.

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

Step 4.1.2

Multiply $2$ by $- 1$ .

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

Step 4.1.3

Multiply $- 1$ by $1$ .

$\frac{- 2 - 1}{4 (- \frac{1}{2}) + 3}$

Step 4.1.4

Subtract $1$ from $- 2$ .

$\frac{- 3}{4 (- \frac{1}{2}) + 3}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 3}{4 (\frac{- 1}{2}) + 3}$

Step 4.2.1.2

Factor $2$ out of $4$ .

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

Step 4.2.1.3

Cancel the common factor.

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

Step 4.2.1.4

Rewrite the expression.

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 4.2.3

Add $- 2$ and $3$ .

$\frac{- 3}{1}$

Step 4.3

Divide $- 3$ by $1$ .

$- 3$