Calculus Examples

$lim_{x \to \frac{3}{2}} 2 x^{3} - 3 x^{2} - 4 x + 2$

Step 1

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

$lim_{x \to \frac{3}{2}} 2 x^{3} - lim_{x \to \frac{3}{2}} 3 x^{2} - lim_{x \to \frac{3}{2}} 4 x + lim_{x \to \frac{3}{2}} 2$

Step 2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to \frac{3}{2}} x^{3} - lim_{x \to \frac{3}{2}} 3 x^{2} - lim_{x \to \frac{3}{2}} 4 x + lim_{x \to \frac{3}{2}} 2$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$2 {(lim_{x \to \frac{3}{2}} x)}^{3} - lim_{x \to \frac{3}{2}} 3 x^{2} - lim_{x \to \frac{3}{2}} 4 x + lim_{x \to \frac{3}{2}} 2$

Step 4

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

$2 {(lim_{x \to \frac{3}{2}} x)}^{3} - 3 lim_{x \to \frac{3}{2}} x^{2} - lim_{x \to \frac{3}{2}} 4 x + lim_{x \to \frac{3}{2}} 2$

Step 5

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

$2 {(lim_{x \to \frac{3}{2}} x)}^{3} - 3 {(lim_{x \to \frac{3}{2}} x)}^{2} - lim_{x \to \frac{3}{2}} 4 x + lim_{x \to \frac{3}{2}} 2$

Step 6

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

$2 {(lim_{x \to \frac{3}{2}} x)}^{3} - 3 {(lim_{x \to \frac{3}{2}} x)}^{2} - 4 lim_{x \to \frac{3}{2}} x + lim_{x \to \frac{3}{2}} 2$

Step 7

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

$2 {(lim_{x \to \frac{3}{2}} x)}^{3} - 3 {(lim_{x \to \frac{3}{2}} x)}^{2} - 4 lim_{x \to \frac{3}{2}} x + 2$

Step 8

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

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Step 8.1

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

$2 {(\frac{3}{2})}^{3} - 3 {(lim_{x \to \frac{3}{2}} x)}^{2} - 4 lim_{x \to \frac{3}{2}} x + 2$

Step 8.2

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

$2 {(\frac{3}{2})}^{3} - 3 {(\frac{3}{2})}^{2} - 4 lim_{x \to \frac{3}{2}} x + 2$

Step 8.3

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Apply the product rule to $\frac{3}{2}$ .

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

Step 9.1.2

Cancel the common factor of $2$ .

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Step 9.1.2.1

Factor $2$ out of $2^{3}$ .

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

Step 9.1.2.2

Cancel the common factor.

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

Step 9.1.2.3

Rewrite the expression.

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

Step 9.1.3

Raise $3$ to the power of $3$ .

$\frac{27}{2^{2}} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 2$

Step 9.1.4

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

$\frac{27}{4} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 2$

Step 9.1.5

Apply the product rule to $\frac{3}{2}$ .

$\frac{27}{4} - 3 \frac{3^{2}}{2^{2}} - 4 (\frac{3}{2}) + 2$

Step 9.1.6

Raise $3$ to the power of $2$ .

$\frac{27}{4} - 3 \frac{9}{2^{2}} - 4 (\frac{3}{2}) + 2$

Step 9.1.7

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

$\frac{27}{4} - 3 (\frac{9}{4}) - 4 (\frac{3}{2}) + 2$

Step 9.1.8

Multiply $- 3 (\frac{9}{4})$ .

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Step 9.1.8.1

Combine $- 3$ and $\frac{9}{4}$ .

$\frac{27}{4} + \frac{- 3 \cdot 9}{4} - 4 (\frac{3}{2}) + 2$

Step 9.1.8.2

Multiply $- 3$ by $9$ .

$\frac{27}{4} + \frac{- 27}{4} - 4 (\frac{3}{2}) + 2$

Step 9.1.9

Move the negative in front of the fraction.

$\frac{27}{4} - \frac{27}{4} - 4 (\frac{3}{2}) + 2$

Step 9.1.10

Cancel the common factor of $2$ .

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Step 9.1.10.1

Factor $2$ out of $- 4$ .

$\frac{27}{4} - \frac{27}{4} + 2 (- 2) \frac{3}{2} + 2$

Step 9.1.10.2

Cancel the common factor.

$\frac{27}{4} - \frac{27}{4} + 2 \cdot - 2 \frac{3}{2} + 2$

Step 9.1.10.3

Rewrite the expression.

$\frac{27}{4} - \frac{27}{4} - 2 \cdot 3 + 2$

Step 9.1.11

Multiply $- 2$ by $3$ .

$\frac{27}{4} - \frac{27}{4} - 6 + 2$

Step 9.2

Combine the numerators over the common denominator.

$- 6 + 2 + \frac{27 - 27}{4}$

Step 9.3

Subtract $27$ from $27$ .

$- 6 + 2 + \frac{0}{4}$

Step 9.4

Find the common denominator.

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Step 9.4.1

Write $- 6$ as a fraction with denominator $1$ .

$\frac{- 6}{1} + 2 + \frac{0}{4}$

Step 9.4.2

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$\frac{- 6}{1} \cdot \frac{4}{4} + 2 + \frac{0}{4}$

Step 9.4.3

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$\frac{- 6 \cdot 4}{4} + 2 + \frac{0}{4}$

Step 9.4.4

Write $2$ as a fraction with denominator $1$ .

$\frac{- 6 \cdot 4}{4} + \frac{2}{1} + \frac{0}{4}$

Step 9.4.5

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$\frac{- 6 \cdot 4}{4} + \frac{2}{1} \cdot \frac{4}{4} + \frac{0}{4}$

Step 9.4.6

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$\frac{- 6 \cdot 4}{4} + \frac{2 \cdot 4}{4} + \frac{0}{4}$

Step 9.5

Combine the numerators over the common denominator.

$\frac{- 6 \cdot 4 + 2 \cdot 4}{4}$

Step 9.6

Simplify each term.

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Step 9.6.1

Multiply $- 6$ by $4$ .

$\frac{- 24 + 2 \cdot 4}{4}$

Step 9.6.2

Multiply $2$ by $4$ .

$\frac{- 24 + 8}{4}$

Step 9.7

Add $- 24$ and $8$ .

$\frac{- 16}{4}$

Step 9.8

Divide $- 16$ by $4$ .

$- 4$