Calculus Examples

$lim_{x \to \frac{1}{2}} \frac{x^{2} - 5 x + 2}{x^{2} + 3 x - 3}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x^{2} - 5 x + 2}{lim_{x \to \frac{1}{2}} x^{2} + 3 x - 3}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x^{2} - lim_{x \to \frac{1}{2}} 5 x + lim_{x \to \frac{1}{2}} 2}{lim_{x \to \frac{1}{2}} x^{2} + 3 x - 3}$

Step 3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{2} - lim_{x \to \frac{1}{2}} 5 x + lim_{x \to \frac{1}{2}} 2}{lim_{x \to \frac{1}{2}} x^{2} + 3 x - 3}$

Step 4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{2} - 5 lim_{x \to \frac{1}{2}} x + lim_{x \to \frac{1}{2}} 2}{lim_{x \to \frac{1}{2}} x^{2} + 3 x - 3}$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{2} - 5 lim_{x \to \frac{1}{2}} x + 2}{lim_{x \to \frac{1}{2}} x^{2} + 3 x - 3}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{2} - 5 lim_{x \to \frac{1}{2}} x + 2}{lim_{x \to \frac{1}{2}} x^{2} + lim_{x \to \frac{1}{2}} 3 x - lim_{x \to \frac{1}{2}} 3}$

Step 7

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{2} - 5 lim_{x \to \frac{1}{2}} x + 2}{{(lim_{x \to \frac{1}{2}} x)}^{2} + lim_{x \to \frac{1}{2}} 3 x - lim_{x \to \frac{1}{2}} 3}$

Step 8

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{2} - 5 lim_{x \to \frac{1}{2}} x + 2}{{(lim_{x \to \frac{1}{2}} x)}^{2} + 3 lim_{x \to \frac{1}{2}} x - lim_{x \to \frac{1}{2}} 3}$

Step 9

Evaluate the limit of $3$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{{(lim_{x \to \frac{1}{2}} x)}^{2} - 5 lim_{x \to \frac{1}{2}} x + 2}{{(lim_{x \to \frac{1}{2}} x)}^{2} + 3 lim_{x \to \frac{1}{2}} x - 1 \cdot 3}$

Step 10

Evaluate the limits by plugging in $\frac{1}{2}$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{{(\frac{1}{2})}^{2} - 5 lim_{x \to \frac{1}{2}} x + 2}{{(lim_{x \to \frac{1}{2}} x)}^{2} + 3 lim_{x \to \frac{1}{2}} x - 1 \cdot 3}$

Step 10.2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{{(\frac{1}{2})}^{2} - 5 (\frac{1}{2}) + 2}{{(lim_{x \to \frac{1}{2}} x)}^{2} + 3 lim_{x \to \frac{1}{2}} x - 1 \cdot 3}$

Step 10.3

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{{(\frac{1}{2})}^{2} - 5 (\frac{1}{2}) + 2}{{(\frac{1}{2})}^{2} + 3 lim_{x \to \frac{1}{2}} x - 1 \cdot 3}$

Step 10.4

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{{(\frac{1}{2})}^{2} - 5 (\frac{1}{2}) + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{\frac{1^{2}}{2^{2}} - 5 (\frac{1}{2}) + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.2

One to any power is one.

$\frac{\frac{1}{2^{2}} - 5 (\frac{1}{2}) + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.3

Raise $2$ to the power of $2$ .

$\frac{\frac{1}{4} - 5 (\frac{1}{2}) + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.4

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{\frac{1}{4} + \frac{- 5}{2} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.5

Move the negative in front of the fraction.

$\frac{\frac{1}{4} - \frac{5}{2} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.6

To write $- \frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{1}{4} - \frac{5}{2} \cdot \frac{2}{2} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.7

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1.7.1

Multiply $\frac{5}{2}$ by $\frac{2}{2}$ .

$\frac{\frac{1}{4} - \frac{5 \cdot 2}{2 \cdot 2} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.7.2

Multiply $2$ by $2$ .

$\frac{\frac{1}{4} - \frac{5 \cdot 2}{4} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.8

Combine the numerators over the common denominator.

$\frac{\frac{1 - 5 \cdot 2}{4} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.9

Simplify the numerator.

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Step 11.1.9.1

Multiply $- 5$ by $2$ .

$\frac{\frac{1 - 10}{4} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.9.2

Subtract $10$ from $1$ .

$\frac{\frac{- 9}{4} + 2}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.10

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

$\frac{\frac{- 9}{4} + 2 \cdot \frac{4}{4}}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.11

Combine $2$ and $\frac{4}{4}$ .

$\frac{\frac{- 9}{4} + \frac{2 \cdot 4}{4}}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.12

Combine the numerators over the common denominator.

$\frac{\frac{- 9 + 2 \cdot 4}{4}}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.13

Simplify the numerator.

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Step 11.1.13.1

Multiply $2$ by $4$ .

$\frac{\frac{- 9 + 8}{4}}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.13.2

Add $- 9$ and $8$ .

$\frac{\frac{- 1}{4}}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.1.14

Move the negative in front of the fraction.

$\frac{- \frac{1}{4}}{{(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{- \frac{1}{4}}{\frac{1^{2}}{2^{2}} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.2.2

One to any power is one.

$\frac{- \frac{1}{4}}{\frac{1}{2^{2}} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.2.3

Raise $2$ to the power of $2$ .

$\frac{- \frac{1}{4}}{\frac{1}{4} + 3 (\frac{1}{2}) - 1 \cdot 3}$

Step 11.2.4

Combine $3$ and $\frac{1}{2}$ .

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

Step 11.2.5

Multiply $- 1$ by $3$ .

$\frac{- \frac{1}{4}}{\frac{1}{4} + \frac{3}{2} - 3}$

Step 11.2.6

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 11.2.7

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.7.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

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

Step 11.2.7.2

Multiply $2$ by $2$ .

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

Step 11.2.8

Combine the numerators over the common denominator.

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

Step 11.2.9

Simplify the numerator.

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Step 11.2.9.1

Multiply $3$ by $2$ .

$\frac{- \frac{1}{4}}{\frac{1 + 6}{4} - 3}$

Step 11.2.9.2

Add $1$ and $6$ .

$\frac{- \frac{1}{4}}{\frac{7}{4} - 3}$

Step 11.2.10

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

$\frac{- \frac{1}{4}}{\frac{7}{4} - 3 \cdot \frac{4}{4}}$

Step 11.2.11

Combine $- 3$ and $\frac{4}{4}$ .

$\frac{- \frac{1}{4}}{\frac{7}{4} + \frac{- 3 \cdot 4}{4}}$

Step 11.2.12

Combine the numerators over the common denominator.

$\frac{- \frac{1}{4}}{\frac{7 - 3 \cdot 4}{4}}$

Step 11.2.13

Simplify the numerator.

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Step 11.2.13.1

Multiply $- 3$ by $4$ .

$\frac{- \frac{1}{4}}{\frac{7 - 12}{4}}$

Step 11.2.13.2

Subtract $12$ from $7$ .

$\frac{- \frac{1}{4}}{\frac{- 5}{4}}$

Step 11.2.14

Move the negative in front of the fraction.

$\frac{- \frac{1}{4}}{- \frac{5}{4}}$

Step 11.3

Dividing two negative values results in a positive value.

$\frac{\frac{1}{4}}{\frac{5}{4}}$

Step 11.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{4} \cdot \frac{4}{5}$

Step 11.5

Cancel the common factor of $4$ .

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Step 11.5.1

Cancel the common factor.

$\frac{1}{4} \cdot \frac{4}{5}$

Step 11.5.2

Rewrite the expression.

$\frac{1}{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{5}$

Decimal Form:

$0.2$