Questions: lim as y approaches 0 of (5y^3 + 8y^2) / (3y^4 - 16y^2)

Solution Steps

To find the limit of the given expression as \( y \) approaches 0, we first simplify the expression if possible. We can factor out the common terms in the numerator and the denominator. Then, we evaluate the limit by substituting \( y = 0 \) if the expression is defined, or use L'Hôpital's Rule if the limit results in an indeterminate form like \( \frac{0}{0} \).

The given expression is

\[ \frac{5y^3 + 8y^2}{3y^4 - 16y^2} \]

We can factor out the common term \( y^2 \) from both the numerator and the denominator:

\[ \frac{y^2(5y + 8)}{y^2(3y^2 - 16)} \]

Cancel the common factor \( y^2 \) from the numerator and the denominator:

\[ \frac{5y + 8}{3y^2 - 16} \]

Now, evaluate the limit of the simplified expression as \( y \) approaches 0:

\[ \lim_{y \to 0} \frac{5y + 8}{3y^2 - 16} \]

Substituting \( y = 0 \) into the expression, we get:

\[ \frac{5(0) + 8}{3(0)^2 - 16} = \frac{8}{-16} = -\frac{1}{2} \]

Final Answer