Calculus Examples

$lim_{y \to 2} \frac{4 y^{3} + 8 y}{y}$

Step 1

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

$\frac{lim_{y \to 2} 4 y^{3} + 8 y}{lim_{y \to 2} y}$

Step 2

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

$\frac{lim_{y \to 2} 4 y^{3} + lim_{y \to 2} 8 y}{lim_{y \to 2} y}$

Step 3

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

$\frac{4 lim_{y \to 2} y^{3} + lim_{y \to 2} 8 y}{lim_{y \to 2} y}$

Step 4

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

$\frac{4 {(lim_{y \to 2} y)}^{3} + lim_{y \to 2} 8 y}{lim_{y \to 2} y}$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $y$ .

$\frac{4 {(lim_{y \to 2} y)}^{3} + 8 lim_{y \to 2} y}{lim_{y \to 2} y}$

Step 6

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

Tap for more steps...

Step 6.1

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

$\frac{4 \cdot 2^{3} + 8 lim_{y \to 2} y}{lim_{y \to 2} y}$

Step 6.2

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

$\frac{4 \cdot 2^{3} + 8 \cdot 2}{lim_{y \to 2} y}$

Step 6.3

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

$\frac{4 \cdot 2^{3} + 8 \cdot 2}{2}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Cancel the common factor of $4 \cdot 2^{3} + 8 \cdot 2$ and $2$ .

Tap for more steps...

Step 7.1.1

Factor $2$ out of $4 \cdot 2^{3}$ .

$\frac{2 (2 \cdot 2^{3}) + 8 \cdot 2}{2}$

Step 7.1.2

Factor $2$ out of $8 \cdot 2$ .

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

Step 7.1.3

Factor $2$ out of $2 (2 \cdot 2^{3}) + 2 (4 \cdot 2)$ .

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

Step 7.1.4

Cancel the common factors.

Tap for more steps...

Step 7.1.4.1

Factor $2$ out of $2$ .

$\frac{2 (2 \cdot 2^{3} + 4 \cdot 2)}{2 (1)}$

Step 7.1.4.2

Cancel the common factor.

$\frac{2 (2 \cdot 2^{3} + 4 \cdot 2)}{2 \cdot 1}$

Step 7.1.4.3

Rewrite the expression.

$\frac{2 \cdot 2^{3} + 4 \cdot 2}{1}$

Step 7.1.4.4

Divide $2 \cdot 2^{3} + 4 \cdot 2$ by $1$ .

$2 \cdot 2^{3} + 4 \cdot 2$

Step 7.2

Simplify each term.

Tap for more steps...

Step 7.2.1

Multiply $2$ by $2^{3}$ by adding the exponents.

Tap for more steps...

Step 7.2.1.1

Multiply $2$ by $2^{3}$ .

Tap for more steps...

Step 7.2.1.1.1

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

$2^{1} \cdot 2^{3} + 4 \cdot 2$

Step 7.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 3} + 4 \cdot 2$

Step 7.2.1.2

Add $1$ and $3$ .

$2^{4} + 4 \cdot 2$

Step 7.2.2

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

$16 + 4 \cdot 2$

Step 7.2.3

Multiply $4$ by $2$ .

$16 + 8$

Step 7.3

Add $16$ and $8$ .

$24$