Calculus Examples

$lim_{y \to 4} \frac{7 y^{2} - 7}{8 y^{2} {(y - 1)}^{3}}$

Step 1

Move the term $\frac{1}{8}$ outside of the limit because it is constant with respect to $y$ .

$\frac{1}{8} lim_{y \to 4} \frac{7 y^{2} - 7}{y^{2} {(y - 1)}^{3}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $4$ .

$\frac{1}{8} \cdot \frac{lim_{y \to 4} 7 y^{2} - 7}{lim_{y \to 4} y^{2} {(y - 1)}^{3}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $4$ .

$\frac{1}{8} \cdot \frac{lim_{y \to 4} 7 y^{2} - lim_{y \to 4} 7}{lim_{y \to 4} y^{2} {(y - 1)}^{3}}$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $y$ .

$\frac{1}{8} \cdot \frac{7 lim_{y \to 4} y^{2} - lim_{y \to 4} 7}{lim_{y \to 4} y^{2} {(y - 1)}^{3}}$

Step 5

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

$\frac{1}{8} \cdot \frac{7 {(lim_{y \to 4} y)}^{2} - lim_{y \to 4} 7}{lim_{y \to 4} y^{2} {(y - 1)}^{3}}$

Step 6

Evaluate the limit of $7$ which is constant as $y$ approaches $4$ .

$\frac{1}{8} \cdot \frac{7 {(lim_{y \to 4} y)}^{2} - 1 \cdot 7}{lim_{y \to 4} y^{2} {(y - 1)}^{3}}$

Step 7

Split the limit using the Product of Limits Rule on the limit as $y$ approaches $4$ .

$\frac{1}{8} \cdot \frac{7 {(lim_{y \to 4} y)}^{2} - 1 \cdot 7}{lim_{y \to 4} y^{2} \cdot lim_{y \to 4} {(y - 1)}^{3}}$

Step 8

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

$\frac{1}{8} \cdot \frac{7 {(lim_{y \to 4} y)}^{2} - 1 \cdot 7}{{(lim_{y \to 4} y)}^{2} \cdot lim_{y \to 4} {(y - 1)}^{3}}$

Step 9

Move the exponent $3$ from ${(y - 1)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{1}{8} \cdot \frac{7 {(lim_{y \to 4} y)}^{2} - 1 \cdot 7}{{(lim_{y \to 4} y)}^{2} \cdot {(lim_{y \to 4} y - 1)}^{3}}$

Step 10

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $4$ .

$\frac{1}{8} \cdot \frac{7 {(lim_{y \to 4} y)}^{2} - 1 \cdot 7}{{(lim_{y \to 4} y)}^{2} \cdot {(lim_{y \to 4} y - lim_{y \to 4} 1)}^{3}}$

Step 11

Evaluate the limit of $1$ which is constant as $y$ approaches $4$ .

$\frac{1}{8} \cdot \frac{7 {(lim_{y \to 4} y)}^{2} - 1 \cdot 7}{{(lim_{y \to 4} y)}^{2} \cdot {(lim_{y \to 4} y - 1 \cdot 1)}^{3}}$

Step 12

Evaluate the limits by plugging in $4$ for all occurrences of $y$ .

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Step 12.1

Evaluate the limit of $y$ by plugging in $4$ for $y$ .

$\frac{1}{8} \cdot \frac{7 \cdot 4^{2} - 1 \cdot 7}{{(lim_{y \to 4} y)}^{2} \cdot {(lim_{y \to 4} y - 1 \cdot 1)}^{3}}$

Step 12.2

Evaluate the limit of $y$ by plugging in $4$ for $y$ .

$\frac{1}{8} \cdot \frac{7 \cdot 4^{2} - 1 \cdot 7}{4^{2} \cdot {(lim_{y \to 4} y - 1 \cdot 1)}^{3}}$

Step 12.3

Evaluate the limit of $y$ by plugging in $4$ for $y$ .

$\frac{1}{8} \cdot \frac{7 \cdot 4^{2} - 1 \cdot 7}{4^{2} \cdot {(4 - 1 \cdot 1)}^{3}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Raise $4$ to the power of $2$ .

$\frac{1}{8} \cdot \frac{7 \cdot 16 - 1 \cdot 7}{4^{2} \cdot {(4 - 1 \cdot 1)}^{3}}$

Step 13.1.2

Multiply $7$ by $16$ .

$\frac{1}{8} \cdot \frac{112 - 1 \cdot 7}{4^{2} \cdot {(4 - 1 \cdot 1)}^{3}}$

Step 13.1.3

Multiply $- 1$ by $7$ .

$\frac{1}{8} \cdot \frac{112 - 7}{4^{2} \cdot {(4 - 1 \cdot 1)}^{3}}$

Step 13.1.4

Subtract $7$ from $112$ .

$\frac{1}{8} \cdot \frac{105}{4^{2} \cdot {(4 - 1 \cdot 1)}^{3}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $- 1$ by $1$ .

$\frac{1}{8} \cdot \frac{105}{4^{2} {(4 - 1)}^{3}}$

Step 13.2.2

Subtract $1$ from $4$ .

$\frac{1}{8} \cdot \frac{105}{4^{2} \cdot 3^{3}}$

Step 13.2.3

Raise $4$ to the power of $2$ .

$\frac{1}{8} \cdot \frac{105}{16 \cdot 3^{3}}$

Step 13.2.4

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

$\frac{1}{8} \cdot \frac{105}{16 \cdot 27}$

Step 13.3

Multiply $16$ by $27$ .

$\frac{1}{8} \cdot \frac{105}{432}$

Step 13.4

Cancel the common factor of $105$ and $432$ .

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Step 13.4.1

Factor $3$ out of $105$ .

$\frac{1}{8} \cdot \frac{3 (35)}{432}$

Step 13.4.2

Cancel the common factors.

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Step 13.4.2.1

Factor $3$ out of $432$ .

$\frac{1}{8} \cdot \frac{3 \cdot 35}{3 \cdot 144}$

Step 13.4.2.2

Cancel the common factor.

$\frac{1}{8} \cdot \frac{3 \cdot 35}{3 \cdot 144}$

Step 13.4.2.3

Rewrite the expression.

$\frac{1}{8} \cdot \frac{35}{144}$

Step 13.5

Multiply $\frac{1}{8} \cdot \frac{35}{144}$ .

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Step 13.5.1

Multiply $\frac{1}{8}$ by $\frac{35}{144}$ .

$\frac{35}{8 \cdot 144}$

Step 13.5.2

Multiply $8$ by $144$ .

$\frac{35}{1152}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{35}{1152}$

Decimal Form:

$0.03038194 \dots$