Calculus Examples

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

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.

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

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 $\frac{a}{2}$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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Step 1.1.2.8.1

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

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

Step 1.1.2.8.2

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

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

Step 1.1.2.8.3

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

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

Step 1.1.2.9

Simplify the answer.

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Step 1.1.2.9.1

Simplify each term.

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Step 1.1.2.9.1.1

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

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

Step 1.1.2.9.1.2

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

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

Step 1.1.2.9.1.3

Cancel the common factor of $4$ .

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Step 1.1.2.9.1.3.1

Cancel the common factor.

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

Step 1.1.2.9.1.3.2

Rewrite the expression.

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

Step 1.1.2.9.1.4

Multiply $- 1$ by $4$ .

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

Step 1.1.2.9.1.5

Cancel the common factor of $2$ .

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Step 1.1.2.9.1.5.1

Factor $2$ out of $- 4 a$ .

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

Step 1.1.2.9.1.5.2

Cancel the common factor.

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

Step 1.1.2.9.1.5.3

Rewrite the expression.

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

Step 1.1.2.9.1.6

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

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

Step 1.1.2.9.1.7

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

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

Step 1.1.2.9.1.8

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

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

Step 1.1.2.9.1.9

Add $1$ and $1$ .

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

Step 1.1.2.9.1.10

Cancel the common factor of $2$ .

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Step 1.1.2.9.1.10.1

Cancel the common factor.

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

Step 1.1.2.9.1.10.2

Rewrite the expression.

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

Step 1.1.2.9.2

Combine the opposite terms in $a^{2} - 2 a^{2} + a + a^{2} - a$ .

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Step 1.1.2.9.2.1

Subtract $a$ from $a$ .

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

Step 1.1.2.9.2.2

Add $a^{2} - 2 a^{2} + a^{2}$ and $0$ .

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

Step 1.1.2.9.3

Subtract $2 a^{2}$ from $a^{2}$ .

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

Step 1.1.2.9.4

Add $- a^{2}$ and $a^{2}$ .

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

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 $\frac{a}{2}$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

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

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

Step 1.1.3.6

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

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

Step 1.1.3.7

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

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Step 1.1.3.7.1

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

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

Step 1.1.3.7.2

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

$\frac{0}{4 {(\frac{a}{2})}^{2} + 2 \frac{a}{2} - a^{2} - a}$

Step 1.1.3.8

Simplify the answer.

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Step 1.1.3.8.1

Simplify each term.

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Step 1.1.3.8.1.1

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

$\frac{0}{4 \frac{a^{2}}{2^{2}} + 2 \frac{a}{2} - a^{2} - a}$

Step 1.1.3.8.1.2

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

$\frac{0}{4 \frac{a^{2}}{4} + 2 \frac{a}{2} - a^{2} - a}$

Step 1.1.3.8.1.3

Cancel the common factor of $4$ .

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Step 1.1.3.8.1.3.1

Cancel the common factor.

$\frac{0}{4 \frac{a^{2}}{4} + 2 \frac{a}{2} - a^{2} - a}$

Step 1.1.3.8.1.3.2

Rewrite the expression.

$\frac{0}{a^{2} + 2 \frac{a}{2} - a^{2} - a}$

Step 1.1.3.8.1.4

Cancel the common factor of $2$ .

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Step 1.1.3.8.1.4.1

Cancel the common factor.

$\frac{0}{a^{2} + 2 \frac{a}{2} - a^{2} - a}$

Step 1.1.3.8.1.4.2

Rewrite the expression.

$\frac{0}{a^{2} + a - a^{2} - a}$

Step 1.1.3.8.2

Combine the opposite terms in $a^{2} + a - a^{2} - a$ .

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Step 1.1.3.8.2.1

Subtract $a^{2}$ from $a^{2}$ .

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

Step 1.1.3.8.2.2

Add $a$ and $0$ .

$\frac{0}{a - a}$

Step 1.1.3.8.2.3

Subtract $a$ from $a$ .

$\frac{0}{0}$

Step 1.1.3.8.2.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.8.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.9

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 1.3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

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

Step 1.3.3.3

Multiply $2$ by $4$ .

$lim_{x \to \frac{a}{2}} \frac{8 x + \frac{d}{d x} [- 4 a x] + \frac{d}{d x} [2 x] + \frac{d}{d x} [a^{2}] + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 4 a x]$ .

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Step 1.3.4.1

Since $- 4 a$ is constant with respect to $x$ , the derivative of $- 4 a x$ with respect to $x$ is $- 4 a \frac{d}{d x} [x]$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a \frac{d}{d x} [x] + \frac{d}{d x} [2 x] + \frac{d}{d x} [a^{2}] + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

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 $n = 1$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a \cdot 1 + \frac{d}{d x} [2 x] + \frac{d}{d x} [a^{2}] + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.4.3

Multiply $- 4$ by $1$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + \frac{d}{d x} [2 x] + \frac{d}{d x} [a^{2}] + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.5

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

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Step 1.3.5.1

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

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + 2 \frac{d}{d x} [x] + \frac{d}{d x} [a^{2}] + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

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 $n = 1$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + 2 \cdot 1 + \frac{d}{d x} [a^{2}] + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.5.3

Multiply $2$ by $1$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + 2 + \frac{d}{d x} [a^{2}] + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.6

Since $a^{2}$ is constant with respect to $x$ , the derivative of $a^{2}$ with respect to $x$ is $0$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + 2 + 0 + \frac{d}{d x} [- a]}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.7

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

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + 2 + 0 + 0}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.8

Simplify.

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Step 1.3.8.1

Combine terms.

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Step 1.3.8.1.1

Add $8 x - 4 a + 2$ and $0$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + 2 + 0}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.8.1.2

Add $8 x - 4 a + 2$ and $0$ .

$lim_{x \to \frac{a}{2}} \frac{8 x - 4 a + 2}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.8.2

Reorder terms.

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{\frac{d}{d x} [4 x^{2} + 2 x - a^{2} - a]}$

Step 1.3.9

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

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- a^{2}] + \frac{d}{d x} [- a]}$

Step 1.3.10

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 1.3.10.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- a^{2}] + \frac{d}{d x} [- a]}$

Step 1.3.10.2

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

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{4 (2 x) + \frac{d}{d x} [2 x] + \frac{d}{d x} [- a^{2}] + \frac{d}{d x} [- a]}$

Step 1.3.10.3

Multiply $2$ by $4$ .

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- a^{2}] + \frac{d}{d x} [- a]}$

Step 1.3.11

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

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Step 1.3.11.1

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

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- a^{2}] + \frac{d}{d x} [- a]}$

Step 1.3.11.2

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

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + 2 \cdot 1 + \frac{d}{d x} [- a^{2}] + \frac{d}{d x} [- a]}$

Step 1.3.11.3

Multiply $2$ by $1$ .

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + 2 + \frac{d}{d x} [- a^{2}] + \frac{d}{d x} [- a]}$

Step 1.3.12

Since $- a^{2}$ is constant with respect to $x$ , the derivative of $- a^{2}$ with respect to $x$ is $0$ .

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + 2 + 0 + \frac{d}{d x} [- a]}$

Step 1.3.13

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

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + 2 + 0 + 0}$

Step 1.3.14

Combine terms.

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Step 1.3.14.1

Add $8 x + 2$ and $0$ .

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + 2 + 0}$

Step 1.3.14.2

Add $8 x + 2$ and $0$ .

$lim_{x \to \frac{a}{2}} \frac{- 4 a + 8 x + 2}{8 x + 2}$

Step 1.4

Cancel the common factor of $- 4 a + 8 x + 2$ and $8 x + 2$ .

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Step 1.4.1

Factor $2$ out of $- 4 a$ .

$lim_{x \to \frac{a}{2}} \frac{2 (- 2 a) + 8 x + 2}{8 x + 2}$

Step 1.4.2

Factor $2$ out of $8 x$ .

$lim_{x \to \frac{a}{2}} \frac{2 (- 2 a) + 2 (4 x) + 2}{8 x + 2}$

Step 1.4.3

Factor $2$ out of $2 (- 2 a) + 2 (4 x)$ .

$lim_{x \to \frac{a}{2}} \frac{2 (- 2 a + 4 x) + 2}{8 x + 2}$

Step 1.4.4

Factor $2$ out of $2$ .

$lim_{x \to \frac{a}{2}} \frac{2 (- 2 a + 4 x) + 2 \cdot 1}{8 x + 2}$

Step 1.4.5

Factor $2$ out of $2 (- 2 a + 4 x) + 2 (1)$ .

$lim_{x \to \frac{a}{2}} \frac{2 (- 2 a + 4 x + 1)}{8 x + 2}$

Step 1.4.6

Cancel the common factors.

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Step 1.4.6.1

Factor $2$ out of $8 x$ .

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

Step 1.4.6.2

Factor $2$ out of $2$ .

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

Step 1.4.6.3

Factor $2$ out of $2 (4 x) + 2 (1)$ .

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

Step 1.4.6.4

Cancel the common factor.

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

Step 1.4.6.5

Rewrite the expression.

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

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 $\frac{a}{2}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

$\frac{- 2 a + 4 \frac{a}{2} + 1}{4 \frac{a}{2} + 1}$

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

Factor $2$ out of $4$ .

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

Step 4.1.1.2

Cancel the common factor.

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

Step 4.1.1.3

Rewrite the expression.

$\frac{- 2 a + 2 a + 1}{4 \frac{a}{2} + 1}$

Step 4.1.2

Add $- 2 a$ and $2 a$ .

$\frac{0 + 1}{4 \frac{a}{2} + 1}$

Step 4.1.3

Add $0$ and $1$ .

$\frac{1}{4 \frac{a}{2} + 1}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2) \frac{a}{2} + 1}$

Step 4.2.2

Cancel the common factor.

$\frac{1}{2 \cdot 2 \frac{a}{2} + 1}$

Step 4.2.3

Rewrite the expression.

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