Calculus Examples

$\frac{lim_{y \to 8} 6 - 7 y^{2}}{3 y^{2} + 4 y}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $8$ .

$\frac{lim_{y \to 8} 6 - lim_{y \to 8} 7 y^{2}}{3 y^{2} + 4 y}$

Step 1.2

Evaluate the limit of $6$ which is constant as $y$ approaches $8$ .

$\frac{6 - lim_{y \to 8} 7 y^{2}}{3 y^{2} + 4 y}$

Step 1.3

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

$\frac{6 - 7 lim_{y \to 8} y^{2}}{3 y^{2} + 4 y}$

Step 1.4

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

$\frac{6 - 7 {(lim_{y \to 8} y)}^{2}}{3 y^{2} + 4 y}$

Step 2

Evaluate the limit of $y$ by plugging in $8$ for $y$ .

$\frac{6 - 7 \cdot 8^{2}}{3 y^{2} + 4 y}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $8$ to the power of $2$ .

$\frac{6 - 7 \cdot 64}{3 y^{2} + 4 y}$

Step 3.1.2

Multiply $- 7$ by $64$ .

$\frac{6 - 448}{3 y^{2} + 4 y}$

Step 3.1.3

Subtract $448$ from $6$ .

$\frac{- 442}{3 y^{2} + 4 y}$

Step 3.2

Factor $y$ out of $3 y^{2} + 4 y$ .

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Step 3.2.1

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

$\frac{- 442}{y (3 y) + 4 y}$

Step 3.2.2

Factor $y$ out of $4 y$ .

$\frac{- 442}{y (3 y) + y \cdot 4}$

Step 3.2.3

Factor $y$ out of $y (3 y) + y \cdot 4$ .

$\frac{- 442}{y (3 y + 4)}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{442}{y (3 y + 4)}$