# Solve for: \lim_{y arrow 0} ((5y^3+8y^2)/(3y^4-16y^2))

## Expression: $\lim_{y \rightarrow 0} \left(\frac{ 5{y}^{3}+8{y}^{2} }{ 3{y}^{4}-16{y}^{2} }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{y \rightarrow 0} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}y} \left( 5{y}^{3}+8{y}^{2} \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}y} \left( 3{y}^{4}-16{y}^{2} \right) }\right)$

Find the derivative

$\lim_{y \rightarrow 0} \left(\frac{ 15{y}^{2}+16y }{ \frac{ \mathrm{d} }{ \mathrm{d}y} \left( 3{y}^{4}-16{y}^{2} \right) }\right)$

Find the derivative

$\lim_{y \rightarrow 0} \left(\frac{ 15{y}^{2}+16y }{ 12{y}^{3}-32y }\right)$

Factor out $y$ from the expression

$\lim_{y \rightarrow 0} \left(\frac{ y \times \left( 15y+16 \right) }{ 12{y}^{3}-32y }\right)$

Factor out $y$ from the expression

$\lim_{y \rightarrow 0} \left(\frac{ y \times \left( 15y+16 \right) }{ y \times \left( 12{y}^{2}-32 \right) }\right)$

Cancel out the common factor $y$

$\lim_{y \rightarrow 0} \left(\frac{ 15y+16 }{ 12{y}^{2}-32 }\right)$

Evaluate the limit

$\frac{ 15 \times 0+16 }{ 12 \times {0}^{2}-32 }$

Simplify the expression

\begin{align*}&-\frac{ 1 }{ 2 } \\&\begin{array} { l }-0.5,& -{2}^{-1}\end{array}\end{align*}

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