Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $h$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Evaluate the limit of $\frac{1}{4}$ which is constant as $h$ approaches $0$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Add $4$ and $0$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Subtract $1$ from $1$ .

$\frac{1}{2} \cdot \frac{0}{4}$

Step 3.4

Divide $0$ by $4$ .

$\frac{1}{2} \cdot 0$

Step 3.5

Multiply $\frac{1}{2}$ by $0$ .

$0$