Calculus Examples

$lim_{x \to \frac{1}{4}} 4 x (x - \frac{3}{8})$

Step 1

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

$4 lim_{x \to \frac{1}{4}} x (x - \frac{3}{8})$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{1}{4}$ .

$4 lim_{x \to \frac{1}{4}} x \cdot lim_{x \to \frac{1}{4}} x - \frac{3}{8}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{4}$ .

$4 lim_{x \to \frac{1}{4}} x \cdot (lim_{x \to \frac{1}{4}} x - lim_{x \to \frac{1}{4}} \frac{3}{8})$

Step 4

Evaluate the limit of $\frac{3}{8}$ which is constant as $x$ approaches $\frac{1}{4}$ .

$4 lim_{x \to \frac{1}{4}} x \cdot (lim_{x \to \frac{1}{4}} x - \frac{3}{8})$

Step 5

Evaluate the limits by plugging in $\frac{1}{4}$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $\frac{1}{4}$ for $x$ .

$4 (\frac{1}{4}) \cdot (lim_{x \to \frac{1}{4}} x - \frac{3}{8})$

Step 5.2

Evaluate the limit of $x$ by plugging in $\frac{1}{4}$ for $x$ .

$4 (\frac{1}{4}) \cdot (\frac{1}{4} - \frac{3}{8})$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $4$ .

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Step 6.1.1

Cancel the common factor.

$4 (\frac{1}{4}) \cdot (\frac{1}{4} - \frac{3}{8})$

Step 6.1.2

Rewrite the expression.

$1 \cdot (\frac{1}{4} - \frac{3}{8})$

Step 6.2

Multiply $\frac{1}{4} - \frac{3}{8}$ by $1$ .

$\frac{1}{4} - \frac{3}{8}$

Step 6.3

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{4} \cdot \frac{2}{2} - \frac{3}{8}$

Step 6.4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.1

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$\frac{2}{4 \cdot 2} - \frac{3}{8}$

Step 6.4.2

Multiply $4$ by $2$ .

$\frac{2}{8} - \frac{3}{8}$

Step 6.5

Combine the numerators over the common denominator.

$\frac{2 - 3}{8}$

Step 6.6

Subtract $3$ from $2$ .

$\frac{- 1}{8}$

Step 6.7

Move the negative in front of the fraction.

$- \frac{1}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{8}$

Decimal Form:

$- 0.125$