Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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 $x$ .

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

Step 5.2

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

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

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.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

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

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

Step 6.3

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.4

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 6.5

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.5.1

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

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

Step 6.5.2

Multiply $4$ by $3$ .

$\frac{3}{12} - \frac{1}{3} \cdot \frac{4}{4}$

Step 6.5.3

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

$\frac{3}{12} - \frac{4}{3 \cdot 4}$

Step 6.5.4

Multiply $3$ by $4$ .

$\frac{3}{12} - \frac{4}{12}$

Step 6.6

Combine the numerators over the common denominator.

$\frac{3 - 4}{12}$

Step 6.7

Subtract $4$ from $3$ .

$\frac{- 1}{12}$

Step 6.8

Move the negative in front of the fraction.

$- \frac{1}{12}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{12}$

Decimal Form:

$- 0.08\overline{3}$