Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

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

Step 16

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 16.1

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

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

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

$\frac{0 + h + 2}{\frac{1}{h} (0 + h - 1 \cdot 4)} - \frac{0 - 1 \cdot 2}{\frac{1}{h} (0 - 1 \cdot 4)}$

Step 17

Simplify the answer.

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Step 17.1

Combine the opposite terms in $\frac{0 + h + 2}{\frac{1}{h} (0 + h - 1 \cdot 4)} - \frac{0 - 1 \cdot 2}{\frac{1}{h} (0 - 1 \cdot 4)}$ .

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Step 17.1.1

Add $0$ and $h$ .

$\frac{h + 2}{\frac{1}{h} (0 + h - 1 \cdot 4)} - \frac{0 - 1 \cdot 2}{\frac{1}{h} (0 - 1 \cdot 4)}$

Step 17.1.2

Add $0$ and $h$ .

$\frac{h + 2}{\frac{1}{h} (h - 1 \cdot 4)} - \frac{0 - 1 \cdot 2}{\frac{1}{h} (0 - 1 \cdot 4)}$

Step 17.1.3

Subtract $1 \cdot 2$ from $0$ .

$\frac{h + 2}{\frac{1}{h} (h - 1 \cdot 4)} - \frac{- 1 \cdot 2}{\frac{1}{h} (0 - 1 \cdot 4)}$

Step 17.1.4

Subtract $1 \cdot 4$ from $0$ .

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

Step 17.2

Simplify each term.

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Step 17.2.1

Multiply $- 1$ by $4$ .

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

Step 17.2.2

Factor $\frac{1}{h}$ out of $\frac{h + 2}{\frac{1}{h} (h - 4)}$ .

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

Step 17.2.3

Combine $h$ and $\frac{h + 2}{h - 4}$ .

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

Step 17.2.4

Dividing two negative values results in a positive value.

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

Step 17.2.5

Cancel the common factor of $2$ and $4$ .

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Step 17.2.5.1

Factor $2$ out of $2$ .

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

Step 17.2.5.2

Cancel the common factors.

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Step 17.2.5.2.1

Factor $2$ out of $\frac{1}{h} \cdot (4)$ .

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

Step 17.2.5.2.2

Cancel the common factor.

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

Step 17.2.5.2.3

Rewrite the expression.

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

Step 17.2.6

Combine $\frac{1}{h}$ and $2$ .

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

Step 17.2.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 17.2.8

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

$\frac{h (h + 2)}{h - 4} - \frac{h}{2}$

Step 17.3

To write $\frac{h (h + 2)}{h - 4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 17.4

To write $- \frac{h}{2}$ as a fraction with a common denominator, multiply by $\frac{h - 4}{h - 4}$ .

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

Step 17.5

Write each expression with a common denominator of $(h - 4) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 17.5.1

Multiply $\frac{h (h + 2)}{h - 4}$ by $\frac{2}{2}$ .

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

Step 17.5.2

Multiply $\frac{h}{2}$ by $\frac{h - 4}{h - 4}$ .

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

Step 17.5.3

Reorder the factors of $(h - 4) \cdot 2$ .

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

Step 17.6

Combine the numerators over the common denominator.

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

Step 17.7

Simplify the numerator.

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Step 17.7.1

Factor $h$ out of $h (h + 2) \cdot 2 - h (h - 4)$ .

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Step 17.7.1.1

Factor $h$ out of $h (h + 2) \cdot 2$ .

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

Step 17.7.1.2

Factor $h$ out of $- h (h - 4)$ .

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

Step 17.7.1.3

Factor $h$ out of $h ((h + 2) \cdot 2) + h (- (h - 4))$ .

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

Step 17.7.2

Apply the distributive property.

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

Step 17.7.3

Move $2$ to the left of $h$ .

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

Step 17.7.4

Multiply $2$ by $2$ .

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

Step 17.7.5

Apply the distributive property.

$\frac{h (2 h + 4 - h - - 4)}{2 (h - 4)}$

Step 17.7.6

Multiply $- 1$ by $- 4$ .

$\frac{h (2 h + 4 - h + 4)}{2 (h - 4)}$

Step 17.7.7

Subtract $h$ from $2 h$ .

$\frac{h (h + 4 + 4)}{2 (h - 4)}$

Step 17.7.8

Add $4$ and $4$ .

$\frac{h (h + 8)}{2 (h - 4)}$