Calculus Examples

$lim_{r \to - \frac{3}{2}} \frac{\sqrt{31 - 12 r^{2}}}{16 r^{3} + 10}$

Step 1

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

$\frac{lim_{r \to - \frac{3}{2}} \sqrt{31 - 12 r^{2}}}{lim_{r \to - \frac{3}{2}} 16 r^{3} + 10}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{r \to - \frac{3}{2}} 31 - 12 r^{2}}}{lim_{r \to - \frac{3}{2}} 16 r^{3} + 10}$

Step 3

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

$\frac{\sqrt{lim_{r \to - \frac{3}{2}} 31 - lim_{r \to - \frac{3}{2}} 12 r^{2}}}{lim_{r \to - \frac{3}{2}} 16 r^{3} + 10}$

Step 4

Evaluate the limit of $31$ which is constant as $r$ approaches $- \frac{3}{2}$ .

$\frac{\sqrt{31 - lim_{r \to - \frac{3}{2}} 12 r^{2}}}{lim_{r \to - \frac{3}{2}} 16 r^{3} + 10}$

Step 5

Move the term $12$ outside of the limit because it is constant with respect to $r$ .

$\frac{\sqrt{31 - 12 lim_{r \to - \frac{3}{2}} r^{2}}}{lim_{r \to - \frac{3}{2}} 16 r^{3} + 10}$

Step 6

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

$\frac{\sqrt{31 - 12 {(lim_{r \to - \frac{3}{2}} r)}^{2}}}{lim_{r \to - \frac{3}{2}} 16 r^{3} + 10}$

Step 7

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

$\frac{\sqrt{31 - 12 {(lim_{r \to - \frac{3}{2}} r)}^{2}}}{lim_{r \to - \frac{3}{2}} 16 r^{3} + lim_{r \to - \frac{3}{2}} 10}$

Step 8

Move the term $16$ outside of the limit because it is constant with respect to $r$ .

$\frac{\sqrt{31 - 12 {(lim_{r \to - \frac{3}{2}} r)}^{2}}}{16 lim_{r \to - \frac{3}{2}} r^{3} + lim_{r \to - \frac{3}{2}} 10}$

Step 9

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

$\frac{\sqrt{31 - 12 {(lim_{r \to - \frac{3}{2}} r)}^{2}}}{16 {(lim_{r \to - \frac{3}{2}} r)}^{3} + lim_{r \to - \frac{3}{2}} 10}$

Step 10

Evaluate the limit of $10$ which is constant as $r$ approaches $- \frac{3}{2}$ .

$\frac{\sqrt{31 - 12 {(lim_{r \to - \frac{3}{2}} r)}^{2}}}{16 {(lim_{r \to - \frac{3}{2}} r)}^{3} + 10}$

Step 11

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

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Step 11.1

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

$\frac{\sqrt{31 - 12 {(- \frac{3}{2})}^{2}}}{16 {(lim_{r \to - \frac{3}{2}} r)}^{3} + 10}$

Step 11.2

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

$\frac{\sqrt{31 - 12 {(- \frac{3}{2})}^{2}}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 12.1.1.1

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

$\frac{\sqrt{31 - 12 ({(- 1)}^{2} {(\frac{3}{2})}^{2})}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.1.2

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

$\frac{\sqrt{31 - 12 ({(- 1)}^{2} \frac{3^{2}}{2^{2}})}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.2

Raise $- 1$ to the power of $2$ .

$\frac{\sqrt{31 - 12 (1 \frac{3^{2}}{2^{2}})}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$\frac{\sqrt{31 - 12 \frac{3^{2}}{2^{2}}}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.4

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

$\frac{\sqrt{31 - 12 \frac{9}{2^{2}}}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.5

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

$\frac{\sqrt{31 - 12 (\frac{9}{4})}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.6

Cancel the common factor of $4$ .

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Step 12.1.6.1

Factor $4$ out of $- 12$ .

$\frac{\sqrt{31 + 4 (- 3) \frac{9}{4}}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.6.2

Cancel the common factor.

$\frac{\sqrt{31 + 4 \cdot - 3 \frac{9}{4}}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.6.3

Rewrite the expression.

$\frac{\sqrt{31 - 3 \cdot 9}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.7

Multiply $- 3$ by $9$ .

$\frac{\sqrt{31 - 27}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.8

Subtract $27$ from $31$ .

$\frac{\sqrt{4}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.9

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}}}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.1.10

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2}{16 {(- \frac{3}{2})}^{3} + 10}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 12.2.1.1

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

$\frac{2}{16 ({(- 1)}^{3} {(\frac{3}{2})}^{3}) + 10}$

Step 12.2.1.2

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

$\frac{2}{16 ({(- 1)}^{3} \frac{3^{3}}{2^{3}}) + 10}$

Step 12.2.2

Raise $- 1$ to the power of $3$ .

$\frac{2}{16 (- \frac{3^{3}}{2^{3}}) + 10}$

Step 12.2.3

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

$\frac{2}{16 (- \frac{27}{2^{3}}) + 10}$

Step 12.2.4

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

$\frac{2}{16 (- \frac{27}{8}) + 10}$

Step 12.2.5

Cancel the common factor of $8$ .

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Step 12.2.5.1

Move the leading negative in $- \frac{27}{8}$ into the numerator.

$\frac{2}{16 (\frac{- 27}{8}) + 10}$

Step 12.2.5.2

Factor $8$ out of $16$ .

$\frac{2}{8 (2) \frac{- 27}{8} + 10}$

Step 12.2.5.3

Cancel the common factor.

$\frac{2}{8 \cdot 2 \frac{- 27}{8} + 10}$

Step 12.2.5.4

Rewrite the expression.

$\frac{2}{2 \cdot - 27 + 10}$

Step 12.2.6

Multiply $2$ by $- 27$ .

$\frac{2}{- 54 + 10}$

Step 12.2.7

Add $- 54$ and $10$ .

$\frac{2}{- 44}$

Step 12.3

Cancel the common factor of $2$ and $- 44$ .

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Step 12.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{- 44}$

Step 12.3.2

Cancel the common factors.

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Step 12.3.2.1

Factor $2$ out of $- 44$ .

$\frac{2 \cdot 1}{2 \cdot - 22}$

Step 12.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot - 22}$

Step 12.3.2.3

Rewrite the expression.

$\frac{1}{- 22}$

Step 12.4

Move the negative in front of the fraction.

$- \frac{1}{22}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{22}$

Decimal Form:

$- 0.0\overline{45}$