Calculus Examples

$lim_{x \to 3} \frac{(\frac{1}{2.9 + 1}) - (\frac{1}{4})}{2.9 - 3}$

Step 1

Evaluate the limit of $\frac{(\frac{1}{2.9 + 1}) - (\frac{1}{4})}{2.9 - 3}$ which is constant as $x$ approaches $3$ .

$\frac{(\frac{1}{2.9 + 1}) - (\frac{1}{4})}{2.9 - 3}$

Step 2

Simplify the answer.

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Step 2.1

Multiply the numerator and denominator of the fraction by $(2.9 + 1) \cdot 4$ .

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Step 2.1.1

Multiply $\frac{\frac{1}{2.9 + 1} - \frac{1}{4}}{2.9 - 3}$ by $\frac{(2.9 + 1) \cdot 4}{(2.9 + 1) \cdot 4}$ .

$\frac{(2.9 + 1) \cdot 4}{(2.9 + 1) \cdot 4} \cdot \frac{\frac{1}{2.9 + 1} - \frac{1}{4}}{2.9 - 3}$

Step 2.1.2

Combine.

$\frac{(2.9 + 1) \cdot 4 (\frac{1}{2.9 + 1} - \frac{1}{4})}{(2.9 + 1) \cdot 4 (2.9 - 3)}$

Step 2.2

Apply the distributive property.

$\frac{(2.9 + 1) \cdot 4 \frac{1}{2.9 + 1} + (2.9 + 1) \cdot 4 (- \frac{1}{4})}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.3

Simplify by cancelling.

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Step 2.3.1

Cancel the common factor of $2.9 + 1$ .

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Step 2.3.1.1

Factor $2.9 + 1$ out of $(2.9 + 1) \cdot 4$ .

$\frac{(2.9 + 1) (4) \frac{1}{2.9 + 1} + (2.9 + 1) \cdot 4 (- \frac{1}{4})}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.3.1.2

Cancel the common factor.

$\frac{(2.9 + 1) \cdot 4 \frac{1}{2.9 + 1} + (2.9 + 1) \cdot 4 (- \frac{1}{4})}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.3.1.3

Rewrite the expression.

$\frac{4 + (2.9 + 1) \cdot 4 (- \frac{1}{4})}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.3.2

Cancel the common factor of $4$ .

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Step 2.3.2.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$\frac{4 + (2.9 + 1) \cdot 4 \frac{- 1}{4}}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.3.2.2

Factor $4$ out of $(2.9 + 1) \cdot 4$ .

$\frac{4 + 4 \cdot (2.9 + 1) \frac{- 1}{4}}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.3.2.3

Cancel the common factor.

$\frac{4 + 4 \cdot (2.9 + 1) \frac{- 1}{4}}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.3.2.4

Rewrite the expression.

$\frac{4 + (2.9 + 1) \cdot - 1}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Add $2.9$ and $1$ .

$\frac{4 + 3.9 \cdot - 1}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.4.2

Multiply $3.9$ by $- 1$ .

$\frac{4 - 3.9}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.4.3

Subtract $3.9$ from $4$ .

$\frac{0.1}{(2.9 + 1) \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

Add $2.9$ and $1$ .

$\frac{0.1}{3.9 \cdot 4 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.5.2

Multiply $3.9 \cdot 4 \cdot 2.9$ .

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Step 2.5.2.1

Multiply $3.9$ by $4$ .

$\frac{0.1}{15.6 \cdot 2.9 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.5.2.2

Multiply $15.6$ by $2.9$ .

$\frac{0.1}{45.24 + (2.9 + 1) \cdot 4 \cdot - 3}$

Step 2.5.3

Add $2.9$ and $1$ .

$\frac{0.1}{45.24 + 3.9 \cdot 4 \cdot - 3}$

Step 2.5.4

Multiply $3.9 \cdot 4 \cdot - 3$ .

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Step 2.5.4.1

Multiply $3.9$ by $4$ .

$\frac{0.1}{45.24 + 15.6 \cdot - 3}$

Step 2.5.4.2

Multiply $15.6$ by $- 3$ .

$\frac{0.1}{45.24 - 46.8}$

Step 2.5.5

Subtract $46.8$ from $45.24$ .

$\frac{0.1}{- 1.56}$

Step 2.6

Divide $0.1$ by $- 1.56$ .

$- 0.0\overline{641025}$