Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{3}{2}$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{3}{2}$ .

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

Step 3

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

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{3}{2}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limits by plugging in $\frac{3}{2}$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $\frac{3}{2}$ for $x$ .

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

Step 7.2

Evaluate the limit of $x$ by plugging in $\frac{3}{2}$ for $x$ .

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

Step 7.3

Evaluate the limit of $x$ by plugging in $\frac{3}{2}$ for $x$ .

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

Step 8

Simplify the answer.

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Step 8.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 8.1.1

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

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

Step 8.1.2

Combine.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Simplify by cancelling.

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Step 8.3.1

Cancel the common factor of $2$ .

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Step 8.3.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

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:

$- 0.94736842 \dots$