Calculus Examples

$lim_{y \to - 2} \frac{4 - 3 y^{2} - y^{3}}{6 - y - y^{2}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $- 2$ .

$\frac{lim_{y \to - 2} 4 - 3 y^{2} - y^{3}}{lim_{y \to - 2} 6 - y - y^{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $- 2$ .

$\frac{lim_{y \to - 2} 4 - lim_{y \to - 2} 3 y^{2} - lim_{y \to - 2} y^{3}}{lim_{y \to - 2} 6 - y - y^{2}}$

Step 3

Evaluate the limit of $4$ which is constant as $y$ approaches $- 2$ .

$\frac{4 - lim_{y \to - 2} 3 y^{2} - lim_{y \to - 2} y^{3}}{lim_{y \to - 2} 6 - y - y^{2}}$

Step 4

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

$\frac{4 - 3 lim_{y \to - 2} y^{2} - lim_{y \to - 2} y^{3}}{lim_{y \to - 2} 6 - y - y^{2}}$

Step 5

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

$\frac{4 - 3 {(lim_{y \to - 2} y)}^{2} - lim_{y \to - 2} y^{3}}{lim_{y \to - 2} 6 - y - y^{2}}$

Step 6

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

$\frac{4 - 3 {(lim_{y \to - 2} y)}^{2} - {(lim_{y \to - 2} y)}^{3}}{lim_{y \to - 2} 6 - y - y^{2}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $- 2$ .

$\frac{4 - 3 {(lim_{y \to - 2} y)}^{2} - {(lim_{y \to - 2} y)}^{3}}{lim_{y \to - 2} 6 - lim_{y \to - 2} y - lim_{y \to - 2} y^{2}}$

Step 8

Evaluate the limit of $6$ which is constant as $y$ approaches $- 2$ .

$\frac{4 - 3 {(lim_{y \to - 2} y)}^{2} - {(lim_{y \to - 2} y)}^{3}}{6 - lim_{y \to - 2} y - lim_{y \to - 2} y^{2}}$

Step 9

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

$\frac{4 - 3 {(lim_{y \to - 2} y)}^{2} - {(lim_{y \to - 2} y)}^{3}}{6 - lim_{y \to - 2} y - {(lim_{y \to - 2} y)}^{2}}$

Step 10

Evaluate the limits by plugging in $- 2$ for all occurrences of $y$ .

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Step 10.1

Evaluate the limit of $y$ by plugging in $- 2$ for $y$ .

$\frac{4 - 3 {(- 2)}^{2} - {(lim_{y \to - 2} y)}^{3}}{6 - lim_{y \to - 2} y - {(lim_{y \to - 2} y)}^{2}}$

Step 10.2

Evaluate the limit of $y$ by plugging in $- 2$ for $y$ .

$\frac{4 - 3 {(- 2)}^{2} - {(- 2)}^{3}}{6 - lim_{y \to - 2} y - {(lim_{y \to - 2} y)}^{2}}$

Step 10.3

Evaluate the limit of $y$ by plugging in $- 2$ for $y$ .

$\frac{4 - 3 {(- 2)}^{2} - {(- 2)}^{3}}{6 + 2 - {(lim_{y \to - 2} y)}^{2}}$

Step 10.4

Evaluate the limit of $y$ by plugging in $- 2$ for $y$ .

$\frac{4 - 3 {(- 2)}^{2} - {(- 2)}^{3}}{6 + 2 - {(- 2)}^{2}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{4 - 3 \cdot 4 - {(- 2)}^{3}}{6 + 2 - {(- 2)}^{2}}$

Step 11.1.2

Multiply $- 3$ by $4$ .

$\frac{4 - 12 - {(- 2)}^{3}}{6 + 2 - {(- 2)}^{2}}$

Step 11.1.3

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

$\frac{4 - 12 - - 8}{6 + 2 - {(- 2)}^{2}}$

Step 11.1.4

Multiply $- 1$ by $- 8$ .

$\frac{4 - 12 + 8}{6 + 2 - {(- 2)}^{2}}$

Step 11.1.5

Subtract $12$ from $4$ .

$\frac{- 8 + 8}{6 + 2 - {(- 2)}^{2}}$

Step 11.1.6

Add $- 8$ and $8$ .

$\frac{0}{6 + 2 - {(- 2)}^{2}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

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

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

Step 11.2.2

Multiply $- 1$ by $4$ .

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

Step 11.2.3

Add $6$ and $2$ .

$\frac{0}{8 - 4}$

Step 11.2.4

Subtract $4$ from $8$ .

$\frac{0}{4}$

Step 11.3

Divide $0$ by $4$ .

$0$