Calculus Examples

lim x \to \frac{3}{2} \frac{- x - 3}{x^{2} + x + 1}

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as

x

$x$ approaches

\frac{3}{2}

$\frac{3}{2}$ .

\frac{lim x \to \frac{3}{2} - x - 3}{lim x \to \frac{3}{2} x^{2} + x + 1}

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

\frac{3}{2}

$\frac{3}{2}$ .

\frac{- lim x \to \frac{3}{2} x - lim x \to \frac{3}{2} 3}{lim x \to \frac{3}{2} x^{2} + x + 1}

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

Step 3

Evaluate the limit of

3

$3$ which is constant as

x

$x$ approaches

\frac{3}{2}

$\frac{3}{2}$ .

\frac{- lim x \to \frac{3}{2} x - 1 \cdot 3}{lim x \to \frac{3}{2} x^{2} + x + 1}

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

\frac{3}{2}

$\frac{3}{2}$ .

\frac{- lim x \to \frac{3}{2} x - 1 \cdot 3}{lim x \to \frac{3}{2} x^{2} + lim x \to \frac{3}{2} x + lim x \to \frac{3}{2} 1}

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

Step 5

Move the exponent

2

$2$ from

x^{2}

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

\frac{- lim x \to \frac{3}{2} x - 1 \cdot 3}{{(lim x \to \frac{3}{2} x)}^{2} + lim x \to \frac{3}{2} x + lim x \to \frac{3}{2} 1}

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

Step 6

Evaluate the limit of

1

$1$ which is constant as

x

$x$ approaches

\frac{3}{2}

$\frac{3}{2}$ .

\frac{- lim x \to \frac{3}{2} x - 1 \cdot 3}{{(lim x \to \frac{3}{2} x)}^{2} + lim x \to \frac{3}{2} x + 1}

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

Step 7

Evaluate the limits by plugging in

\frac{3}{2}

$\frac{3}{2}$ for all occurrences of

x

$x$ .

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Step 7.1

Evaluate the limit of

x

$x$ by plugging in

\frac{3}{2}

$\frac{3}{2}$ for

x

$x$ .

\frac{- \frac{3}{2} - 1 \cdot 3}{{(lim x \to \frac{3}{2} x)}^{2} + lim x \to \frac{3}{2} x + 1}

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

Step 7.2

Evaluate the limit of

x

$x$ by plugging in

\frac{3}{2}

$\frac{3}{2}$ for

x

$x$ .

\frac{- \frac{3}{2} - 1 \cdot 3}{{(\frac{3}{2})}^{2} + lim x \to \frac{3}{2} x + 1}

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

Step 7.3

Evaluate the limit of

x

$x$ by plugging in

\frac{3}{2}

$\frac{3}{2}$ for

x

$x$ .

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

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

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

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

Step 8

Simplify the answer.

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Step 8.1

Multiply the numerator and denominator of the fraction by

2

$2$ .

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Step 8.1.1

Multiply

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

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

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 8.1.2

Combine.

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

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

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

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

Step 8.2

Apply the distributive property.

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

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

Step 8.3

Simplify by cancelling.

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Step 8.3.1

Cancel the common factor of

2

$2$ .

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Step 8.3.1.1

Move the leading negative in

- \frac{3}{2}

$- \frac{3}{2}$ into the numerator.

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

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

Step 8.3.1.2

Cancel the common factor.

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

Step 8.3.1.3

Rewrite the expression.

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

Step 8.3.2

Cancel the common factor of

$2$ .

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Step 8.3.2.1

Cancel the common factor.

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

Step 8.3.2.2

Rewrite the expression.

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

Step 8.4

Simplify the numerator.

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Step 8.4.1

Multiply

$2 (- 1 \cdot 3)$ .

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Step 8.4.1.1

Multiply

$- 1$ by

$3$ .

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

Step 8.4.1.2

Multiply

$2$ by

$- 3$ .

$\frac{- 3 - 6}{2 {(\frac{3}{2})}^{2} + 3 + 2 \cdot 1}$

Step 8.4.2

Subtract

$6$ from

$- 3$ .

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

Step 8.5

Simplify the denominator.

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Step 8.5.1

Apply the product rule to

$\frac{3}{2}$ .

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

Step 8.5.2

Cancel the common factor of

$2$ .

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Step 8.5.2.1

Factor

$2$ out of

$2^{2}$ .

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

Step 8.5.2.2

Cancel the common factor.

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

Step 8.5.2.3

Rewrite the expression.

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

Step 8.5.3

Raise

$3$ to the power of

$2$ .

$\frac{- 9}{\frac{9}{2} + 3 + 2 \cdot 1}$

Step 8.5.4

Multiply

$2$ by

$1$ .

$\frac{- 9}{\frac{9}{2} + 3 + 2}$

Step 8.5.5

To write

$3$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\frac{- 9}{\frac{9}{2} + 3 \cdot \frac{2}{2} + 2}$

Step 8.5.6

Combine

$3$ and

$\frac{2}{2}$ .

$\frac{- 9}{\frac{9}{2} + \frac{3 \cdot 2}{2} + 2}$

Step 8.5.7

Combine the numerators over the common denominator.

$\frac{- 9}{\frac{9 + 3 \cdot 2}{2} + 2}$

Step 8.5.8

Simplify the numerator.

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Step 8.5.8.1

Multiply

$3$ by

$2$ .

$\frac{- 9}{\frac{9 + 6}{2} + 2}$

Step 8.5.8.2

Add

$9$ and

$6$ .

$\frac{- 9}{\frac{15}{2} + 2}$

Step 8.5.9

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\frac{- 9}{\frac{15}{2} + 2 \cdot \frac{2}{2}}$

Step 8.5.10

Combine

$2$ and

$\frac{2}{2}$ .

$\frac{- 9}{\frac{15}{2} + \frac{2 \cdot 2}{2}}$

Step 8.5.11

Combine the numerators over the common denominator.

$\frac{- 9}{\frac{15 + 2 \cdot 2}{2}}$

Step 8.5.12

Simplify the numerator.

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Step 8.5.12.1

Multiply

$2$ by

$2$ .

$\frac{- 9}{\frac{15 + 4}{2}}$

Step 8.5.12.2

Add

$15$ and

$4$ .

$\frac{- 9}{\frac{19}{2}}$

Step 8.6

Multiply the numerator by the reciprocal of the denominator.

$- 9 (\frac{2}{19})$

Step 8.7

Multiply

$- 9 (\frac{2}{19})$ .

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Step 8.7.1

Combine

$- 9$ and

$\frac{2}{19}$ .

$\frac{- 9 \cdot 2}{19}$

Step 8.7.2

Multiply

$- 9$ by

$2$ .

$\frac{- 18}{19}$

Step 8.8

Move the negative in front of the fraction.

$- \frac{18}{19}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{18}{19}$

Decimal Form:

$- 0.94736842 \dots$