Precalculus Examples

$\frac{lim_{y \to 8} 3 - 5 y^{4}}{y^{7} - 2 y^{3} + 1}$

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 $8$ .

$\frac{lim_{y \to 8} 3 - lim_{y \to 8} 5 y^{4}}{y^{7} - 2 y^{3} + 1}$

Step 1.2

Evaluate the limit of $3$ which is constant as $y$ approaches $8$ .

$\frac{3 - lim_{y \to 8} 5 y^{4}}{y^{7} - 2 y^{3} + 1}$

Step 1.3

Move the term $5$ outside of the limit because it is constant with respect to $y$ .

$\frac{3 - 5 lim_{y \to 8} y^{4}}{y^{7} - 2 y^{3} + 1}$

Step 1.4

Move the exponent $4$ from $y^{4}$ outside the limit using the Limits Power Rule.

$\frac{3 - 5 {(lim_{y \to 8} y)}^{4}}{y^{7} - 2 y^{3} + 1}$

Step 2

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

$\frac{3 - 5 \cdot 8^{4}}{y^{7} - 2 y^{3} + 1}$

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 $4$ .

$\frac{3 - 5 \cdot 4096}{y^{7} - 2 y^{3} + 1}$

Step 3.1.2

Multiply $- 5$ by $4096$ .

$\frac{3 - 20480}{y^{7} - 2 y^{3} + 1}$

Step 3.1.3

Subtract $20480$ from $3$ .

$\frac{- 20477}{y^{7} - 2 y^{3} + 1}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{20477}{y^{7} - 2 y^{3} + 1}$